Calculator Instructions · © 2012 Carnegie Learning Calculator Instructions C-1 Calculator...

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Calculator Instructions C-1

Calculator Instructions

1.3 Graphing a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3

1.3 Graphing a Piecewise Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-3

2.2 Completing a Table of Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4

2.2 Using the Value Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-4

2.2 Using the Intersect Feature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-5

3.1 Graphing a Function Using a Table of Values . . . . . . . . . . . . . . . . . . . . . . . . C-6

3.1 Determining a Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-7

4.3 Generating Two Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8

7.2 Graphing a System of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . C-8

8.2 Constructing a Box-and-Whisker Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9

8.4 Determining Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-9

9.4 Comparing a Data Set With Its Linear Regression . . . . . . . . . . . . . . . . . . . . C-10

11.1 Identifying the Absolute Maximum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11

11.3 Determining Zeros . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11

12.2 Verifying the Product of Two Binomials . . . . . . . . . . . . . . . . . . . . . . . . . . . C-11

13.2 Using the CBR With Your Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-12

13.2 Determining a Quadratic Regression Equation . . . . . . . . . . . . . . . . . . . . . . C-13

13.2 Graphing Quadratics Using the CBR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-14

15.2 Graphing a Step Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-15

15.2 Graphing a Greatest Integer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-15

15.2 Graphing a Least Integer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-16

15.3 Graphing an Inverse of a Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C-16

16.1 Determining an Exponential Regression Equation . . . . . . . . . . . . . . . . . . . C-16

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Calcu

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Calculator Instructions

You can use a graphing calculator to graph

a function.

Step 1: Press Y=. Your cursor should be

blinking on the line \Y1=. Enter

the equation. To enter a variable

like x, press the key with

X, T, Ø, n once.

Step 2: Press WINDOW to set the

bounds and intervals you

want displayed.

Step 3: Press GRAPH to view the graph.


piecewise functions.

Step 1: Press Y=. Enter the first section of the

function within parentheses. Then press

the division button.

Step 2: Press the ( key twice and enter the first part

of the compound inequality within

parentheses.

Step 3: Enter the second part of the compound

inequality within parentheses and then

type two closing parentheses.

Press GRAPH here to see the first section of

the piecewise function.

Step 4: Enter the remaining sections

of the piecewise functions as

Y2 and Y3.

Graphing a Function1.3

Graphing a Piecewise Function1.3


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C-4 Calculator

Cal

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You can use a graphing calculator to complete a

table of values for a given function.

Step 1: Press Y=

Step 2: Enter the function. Press ENTER.

Step 3: Press 2ND TBLSET (above WINDOW).

TblStart is the starting data value

for your table. Enter this value.

ΔTbl (read “delta table”) is the

increment. This value tells the table

what intervals to count by for the

independent quantity. If ΔTbl = 1 then the

values in your table would go up by 1s. If

ΔTbl = -1, the values would go down by 1s.

Enter the ΔTbl.

Step 4: Press 2ND TABLE (above GRAPH). Use the up and

down arrows to scroll through the data.

You can use the value feature on a graphing

calculator to determine an exact data value

on a graph.

Step 1: Press Y=. Enter your function.

Step 2: Press WINDOW. Set appropriate values for

your function. Then press GRAPH.

Step 3: Press 2ND and then CALC. Select 1:value.Press

ENTER. Then type the given independent value

next to X= and press ENTER. The cursor moves

to the given independent value and the

corresponding dependent value is displayed at

the bottom of the screen.

Completing a Table of Values2.2

Using the Value Feature2.2


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Calculator C-5

Calcu

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You can use the intersect feature to determine

an independent value when given a dependent

value.

Step 1: Press Y5. Enter the two equations, one next

to Y15 and one next to Y25.

Step 2: Press WINDOW. Set appropriate bounds so you

can see the intersection of the two

equations. Then press GRAPH.

Step 3: Press 2ND CALC and then select 5:intersect.

The cursor should appear somewhere on

one of the graphs, and at the bottom

of the screen you will see First curve?

Press ENTER.

The cursor should then move to somewhere on

the other graph, and you will see Second

curve? Press ENTER.

You will see Guess? at the bottom of the

screen. Move the cursor to where you think

the intersection point is and Press ENTER.

The intersection point will appear.

Using the Intersect Feature2.2


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C-6 Calculator Instructions

Cal

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You can use a graphing calculator to represent a

data set.

Step 1: Press STAT and then press ENTER to select

1:Edit. In the L1 column, enter the independent

quantity values by typing each value followed

by ENTER.

Step 2: Use the right arrow key to move to the L2

column. Enter the dependent quantity values.

Step 3: Press 2ND and STAT PLOT. Press 4

to turn off any plots. Press

ENTER. Then press 2ND and STAT

PLOT again. Press ENTER to access

the information about Plot 1. The

cursor should be on the word On.

Press ENTER to turn on Plot 1.

Step 4: Use the arrow keys to move down to Xlist.

Press 2ND L1 to set your L1 values as your

x-values. Scroll to Ylist and Press 2ND L2

to set your L2 values as your y-values.

Step 5: Press WINDOW to set the bounds of your graph.

Press GRAPH to create a graph of the data.

Step 6: Use the TRACE feature and the left and right

arrow keys to move between the points on the

plot.

Graphing a Function Using a Table of Values3.1


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You can use a graphing calculator to determine

the linear regression equation of a data set.

Step 1: Diagnostics must be turned on so that all

needed data is displayed. Press 2nd CATALOG to

display the catalog. Scroll to DiagnosticOn

and press ENTER. Then press ENTER again. The

calculator should display the word Done.

Step 2: Press STAT and use the right arrow key to

show the CALC menu. Type 4 to choose

LinReg(ax1b) and press ENTER.

Step 3: Make sure L1 is listed next to Xlist and L2

is listed next to Ylist. Scroll down to

Calculate and press ENTER.

The calculator should show y = ax + b

as well as four values labeled a, b, r2,

and r.

Step 4: Press Y=. Enter the linear regression

equation next to Y1=. Then press GRAPH

to see the line of best fit.

The calculator will automatically copy the

linear regression equation to Y1 if you enter

LinReg(ax + b) Y1. Repeat Step 2 to enter

LinReg(ax + b), then press VARS and use the

right arrow keys to show Y-VARS. Press

FUNCTION and select Y1. When you press ENTER

the equation will appear in Y1.

Determining a Linear Regression3.1


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Cal

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You can use a graphing calculator to generate

two sequences at the same time in order to

determine a certain term in a sequence.

Step 1: Within a set of brackets, enter the first

term number followed by a comma and then the

first term value of the sequence. The 2ND key

is used to enter the brackets. Press ENTER.

Step 2: Provide direction to the calculator on how to

generate each term of the sequence.

Press 2ND{2NDANS(1) and then indicate how the

term numbers will increase or decrease, and

by how much by entering the plus or minus

sign and the amount of increase or decrease.

Then press ,2ND ANS(2) and enter the common

difference of the term values. Then close the

brackets by pressing 2ND} and press ENTER.

The calculator will display the next term

number and value.

Step 3: Press ENTER and the next term number

and value will be displayed.

Step 4: Continue pressing ENTER until you reach the

nth term number and value you want to

determine.


a system of linear inequalities.

Step 1: Press Y= and enter the two inequalities

as Y1 and Y2.

Step 2: While still in the Y= window, access the

inequality function by moving your cursor

to the left until the \ flashes. Press ENTER

to select the appropriate inequality

symbol ( or ).

Step 3: Press WINDOW and set the bounds.

Step 4: Press GRAPH.

Generating Two Sequences4.3

Graphing a System of Linear Inequalities7.2


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Calcu

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You can use a graphing calculator to construct a

box-and-whisker plot.

Step 1: Press STAT and then press ENTER to select

1:Edit.

Step 2: Enter the data values of the data set in List 1.

Step 3: Press 2nd and STAT PLOT, which is above the

Y= button.

Step 4: Select 1: and press ENTER. Then highlight

PLOT 1 and press ENTER to turn Plot 1 on.

Then scroll down to Type: and select the

box-and-whisker icon. Press ENTER.

Step 5: Make sure the XList is using the correct

list. Then press GRAPH.


the standard deviation of a data set.

Step 1: Press STAT and then ENTER to select 1:Edit.

Enter each data set into its own List.

Step 2: Press STAT then scroll to the right to

highlight CALC. Press ENTER to select 1:

Var-Stats. Press ENTER.

Step 3: Your screen should display 1—Var Stats. Press

2ND then the list you want the calculator to

use for these calculations.

Step 4: Your calculator should display the same data

values as when you determined the mean.

However, this time use the value for sx.

Constructing a Box-and-Whisker Plot8.2

Determining Standard Deviation8.4


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Cal

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You can use a graphing calculator to show how

the actual values of a data set differ from the

values predicted by a linear regression.

Step 1: Enter the data values, press STAT, select

CALC, and then select 4:LinReg(ax+b). Scroll

down to Store RegEQ: Press VARS, select

Y-VARS at the top, and then press 1 two

times. Then select Calculate.

Step 2: Press STAT and then 1. Then press the right

arrow key until you get to L6. Press the up

arrow key and then the right arrow key.

Step 3: If the list of residuals is not already

displayed, press 2ND and then LIST. Select

7↓RESID. Press ENTER.

Step 4: Press 2ND, STAT PLOT, 1 to turn on the plot

and choose the type of display for the graph.

Press ZOOM and then 9 to show the data and

the line of best fit.

You can also use a graphing calculator to graph a

residual plot.

Step 5: Press STAT and then 1. Copy the data from the

residuals list to L6. You can round the data

values if you wish.

Step 6: Press 2ND, STAT PLOT, and then 1. Make sure

L1 is entered next to Xlist and L6 is entered

next to Ylist.

Step 7: Press STAT, select CALC, and then select

2:2-Var Stats. Make sure L1 is entered next

to Xlist and L6 is entered next to Ylist.

Select Calculate and then press ZOOM, 9 to

see the residual plot.

Comparing a Data Set With Its Linear Regression9.4


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You can use a graphing calculator to identify

the absolute maximum of a quadratic function.

Step 1: Enter the function and press GRAPH.

Step 2: Press 2ND and then CALC.

Select 4: maximum.

Step 3: Move your cursor to any point on the

curve where the graph is increasing

and press ENTER.

Step 4: Move your cursor to a point directly

opposite the point in Step 3 on the curve

where the graph is decreasing and press

ENTER. Then press ENTER one more time.


the zeros of a quadratic function.

Step 1: Press 2ND and then CALC. Select 2: zero.

Step 2: Determine the left and right bounds for each

point that appears to be a zero. Then press

ENTER.

You can use a graphing calculator to check if

the product of two binomials is correct.

Step 1: Press Y5. Enter the two binomials multiplied

next to Y1. Then enter their product next to Y2.

To distinguish between the graphs of Y1 and Y2,

move your cursor to the left of Y2 until the \

flashes. Press ENTER one time to select the

bold \.

Step 2: Press WINDOW to set the bounds and intervals

for the graph.


Identifying the Absolute Maximum11.1

Determining Zeros11.3

Verifying the Product of Two Binomials12.2


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Cal

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Step 1: Attach the CBR to your graphing calculator with the unit-to-unit

link cable that comes with the CBR.

Step 2: Press APPS and choose CBL/CBR.

Step 3: Press any key to advance past the introduction

screen.

Step 4: Select 3:Ranger.

Step 5: Press ENTER to advance past the introduction

screen.

Step 6: Select 3:APPLICATIONS.

Step 7: For this experiment, we will be using customary

units of measure. Select 2:FEET.

Step 8: Open the CBR. Make sure that the indicator below

the CBR scanner (that looks like a metallic metal

mesh circle) is selected on the ball icon.

Step 9: Select 3:BALL BOUNCE. Follow the instructions

with the BALL BOUNCE program.

Step 10: At this point, you can detach the CBR from the

calculator. Press ENTER.

Step 11: Press the TRIGGER when you are ready to record the data. You

should hear a faint tapping sound from the CBR when it is

recording data.

Using the CBR With Your Calculator13.2


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Calcu

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the quadratic regression of a set of data.

Step 1: Press ENTER to return to the Plot Menu.

Scroll down to 7:QUIT. Press ENTER.

Step 2: You will see on the screen Lists 1 through

4(L1-L4). The data needed to determine

the quadratic regression is located in

Lists 1 and 2 which measures time and

distance.

Step 3: Press STAT. Scroll right to CALC. Choose

5:QuadReg. Press ENTER.

Step 4: Select L1 , L2. Press ENTER.

Step 5: The information shows you the standard form

of a quadratic equation and the values of a,

b, c, and r2.

Determining a Quadratic Regression Equation13.2


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Cal

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Step 1: Use the built-in SelectO feature of the

TI-83 to select the first “good” toss

that is collected.

Step 2: With the plot displayed, press 2nd LlST

Scroll to OPS, and then chose 8:select(

to enter where you want to store the

selected data. To use L3 and L4, press

2nd L3 , 2nd L4 ) ENTER.

Step 3: To actually select a part of the graph

you will use, press c to move to the left

end of the data you want to keep. Press

ENTER. This sets the left bound.

Press c to move to the right end of the

data you want. Press ENTER. The selected

data will be placed in L3 and L4, and

then this data will be displayed.

Graphing Quadratics Using the CBR13.2


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Calcu

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You can use a graphing calculator to graph a

greatest integer function, G(x) 5 ⎣x⎦.

Step 1: Press MODE. Ensure that the mode is set on

Dot. If not, navigate down to Dot and press

enter. Dot should now be highlighted.

Step 2: Press Y5, and then press MATH.

Step 3: Scroll over to the NUM menu. Press 5 to

select 5: int( .

Step 4: Press X,T,Q,n and then type the closing

parenthesis.


Graphing a Greatest Integer Function15.2

Graphing a Step Function15.2

You can use a graphing calculator to graph a

step function.

Step 1: Press Y5. Enter the first interval and its

bounds in Y1=, just as you did for piecewise

functions. This represents the first “step”

of the step function.

Step 2: Scroll down to Y25. Enter the second interval

and its bounds to represent the second “step”

of the step function.

Step 3: Follow the same method to enter the remaining

“steps” of the function. Then press GRAPH.


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Cal

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Step 1: Graph f(x), f21(x), and the line y 5 x

together on the grid shown.

Step 2: Heavily trace the graph of f(x) with

a pencil.

Step 3: Fold the graph along the line y 5 x, and rub

the paper so that the image of the graph of

f(x) appears.


the exponential regression equation for a data

set.

Step 1: Press STAT and select 1:Edit. Enter the data

set with the independent variable in L1 and

the dependent variable in L2.

Step 2: Press STAT and scroll to CALC. Then scroll

down to 0:ExpReg. Press ENTER twice.

The calculator will display the values of each

variable in the form y 5 a.bx.

Step 3: The r-value displayed represents the

correlation coefficient.

Graphing an Inverse of a Function15.3

Determining an Exponential Regression Equation16.1

You can use a graphing calculator to graph the

least integer function L(x) = ⎡x⎤.

Step 1: Press Y5. Press the negative sign (2)and then

press MATH.

Step 2: Scroll over to the NUM menu and press 5 to

select 5: int( .

Step 3: Press the negative sign (2)and then press

X,T,Q,n. Next, type the closing parenthesis.


Graphing a Least Integer Function15.2


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Glossary G-1

A

absolute maximumA function has an absolute maximum if there is a point that has a y-coordinate that is greater than the y-coordinates of every other point on the graph.

Example

The ordered pair (4, 2) is the absolute maximum of the graph of the function f(x) 5 2 1 __

2 x2 1 4x 2 6.

28 26 24 22

4

2

222

(4, 2)

0 4 6 8x

24

26

28

y

y 5 212 x2 1 4x 2 6

8

6

absolute minimumA function has an absolute minimum if there is a point that has a y-coordinate that is less than the y-coordinates of every other point on the graph.

Example

The ordered pair (1, 24) is the absolute minimum of the graph of the function y 5 2 __

3 x2 2 4 __

3 x 2 10 ___

3 .

28 26 24 22

4

2

222

(1, 24)

0 4 6 8x

24

26

28

y

23y 5 x2 2

43

103x 2

8

6

argument of a functionThe argument of a function is the variable on which the function operates.

Example

In the function f(x 1 5) 5 32, the argument is x 1 5.

arithmetic sequenceAn arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is a constant.

Example

The sequence 1, 3, 5, 7 is an arithmetic sequence with a common difference of 2.

Glossary

Glossary

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G-2 Glossary

axis of symmetryThe axis of symmetry of a parabola is the vertical line that passes through the vertex and divides the parabola into two mirror images.

Example

Line K is the axis of symmetry of this parabola.

K

B

basic functionA basic function is the simplest function of its type.

Examples

The basic linear function is f(x) 5 x. The basic exponential function is g(x) 5 2x. The basic quadratic function is h(x) 5 x2.

binThe width of a bar in a histogram represents an interval of data and is often referred to as a bin.

binomialPolynomials with exactly two terms are binomials.

Example

The polynomial 3x 1 5 is a binomial.

box-and-whisker plotA box-and-whisker plot displays a data distribution based on a five number summary.

Example

The box-and-whisker plots compare the test scores from two algebra classes.

Class 1

0 10 20 30 40 50

48 60 72 84 94

57 65 8085 95

60 70 80 90 100

Class 2

break-even pointThe break-even point is the point where two functions are equal.

Example

x4540

300

350

400

C(g) 5 5.77g + 45

I(g) = 8.5g450

20 25Gearboxes

Dol

lars

200

250

353015

150

10

100

5

50

0

ybreak-even point

C

categorical dataData that can be grouped into categories are called categorical data.

causationCausation is when one event causes a second event.

Glo

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Glossary G-3

closed (closure)When an operation is performed on any of the numbers in a set and the result is a number that is also in the same set, the set is said to be closed (or to have closure) under that operation.

Example

The set of whole numbers is closed under addition. The sum of any two whole numbers is always another whole number.

closed intervalA closed interval [a, b] describes the set of all numbers between a and b, including a and b.

A half-closed or half-open interval (a, b] describes the set of all numbers between a and b, including b but not including a. Or, [a, b) describes the set of all numbers between a and b, including a but not including b.

coefficientWithin a polynomial, a coefficient is a number multiplied by a power.

Example

The term 3x5 has a coefficient of 3.

coefficient of determinationThe coefficient of determination measures the “strength” of the relationship between the original data and its quadratic regression equation.

common differenceThe difference between any two consecutive terms in an arithmetic sequence is called the common difference. It is typically represented by the variable d.

Example

The sequence 1, 3, 5, 7 is an arithmetic sequence with a common difference of 2.

common ratioThe ratio between any two consecutive terms in a geometric sequence is called the common ratio. It is typically represented by the variable r.

Example

The sequence 2, 4, 8, 16 is a geometric sequence with a common ratio of 2.

common responseA common response is when a variable other than the ones measured cause the same result as the one observed in the experiment.

completing the squareCompleting the square is a process for writing a quadratic expression in vertex form which then allows you to solve for the zeros.

complex numbersThe set of complex numbers is the set of all numbers written in the form a 1 bi, where a and b are real numbers.

composition of functionsA composition of functions is the combination of functions such that the output from one function becomes the input for the next function.

Example

The composition of function f(x) composed with g(x) is denoted (f g)(x) or f(g(x)). It is read as “f composed with g(x)” or “f of g(x).”

compound inequalityA compound inequality is an inequality that is formed by the union, “or,” or the intersection, “and,” of two simple inequalities.

Example

The statement “x . 5 or x , 2 5 is a compound inequality.

compound interestIn a compound interest account, the interest earned at the end of each year is a percent of the account balance at the beginning of the year.

Example

Sonya opens a savings account with $100. She earns $4 in interest the first year. The compound interest y is found by using the equation y 5 100(1 1 0.04)t, where t is the time in years.

confounding variableA confounding variable is when there are other variables in an experiment that are unknown or unobserved.

Glossary


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G-4 Glossary

conjunctionA compound inequality in the form a , x , b, where a and b are any real numbers, is a conjunction.

Example

The compound inequality “x # 1 and x . 2 3” is a conjunction.

consistent systemsSystems that have one or many solutions are called consistent systems.

constant functionIf the dependent variable of a function does not change or remains constant over the entire domain, then the function is called a constant function.

Example

The function shown is a constant function.

8

6

4

2

0

22

24

26

28

21020 4 6 8 10 12 14 16 18

x

y

constraintsIn a system of linear inequalities, the inequalities are known as constraints because the values of the expressions are “constrained” to lie within a certain region on the graph.

continuous dataContinuous data are data which can take any numerical value within a range.

coordinate notationCoordinate notation is a notation that uses ordered pairs to describe a transformation in a coordinate plane.

Example

The coordinate notation (x, y) → (x 1 a, y), where a is a real number, indicates a horizontal translation.

correlation coefficientThe correlation coefficient indicates how closely the data points form a straight line.

Example

The correlation coefficient for these data is 20.9935. The value is negative because the equation has a negative slope. The value is close to 21 because the data are very close to forming a straight line.

108

96

84

72

60

48

Per

cent

of S

culp

ture

Rem

aini

ng

36

24

12

010 2 3 4 5 6

Time (hours)

How Long Does an IceSwan Last?

7 8 9x

y

counterexampleA counterexample is a single example that shows that a statement is not true.

Example

Your friend claims that you add fractions by adding the numerators and then adding the denominators. A counterexample is 1 __

2 1 1 __

2 . The sum of these two

fractions is 1. Your friend’s method results in 2 __ 4 , or 1 __

2 .

Your friend’s method is incorrect.

Glo

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Glossary G-5

Glossary

dependent quantityWhen one quantity depends on another in a problem situation, it is said to be the dependent quantity.

Example

In the relationship between driving time and distance traveled, distance is the dependent quantity, because distance depends on the driving time.

difference of two cubesThe difference of two cubes is an expression in the form a3 2 b3 that can be factored as (a 2 b)(a2 1 ab 1 b2).

difference of two squaresThe difference of two squares is an expression in the form a2 2 b2 that can be factored as (a 1 b)(a 2 b).

dilation factorThe dilation factor is the common factor which every y-coordinate of a graph is multiplied by to produce a vertical dilation.

discrete dataDiscrete data are data that has only a finite number of values or data that can be “counted.”

discriminantThe discriminant is the radicand expression in the Quadratic Formula which “discriminates” the number of roots of a quadratic equation.

Example

The discriminant in the Quadratic Formula is the expression b2 2 4ac.

disjunctionA compound inequality in the form x , a or x . b, where a and b are any real numbers, is a disjunction.

Example

The compound inequality “x , 22 or x . 1” is a disjunction.

cube rootYou can say that a number b is a cube root of a number a if b3 5 a.

Example

The cube root of 8 is 2.

D

data distributionA data distribution is the way in which data are spread out or clustered together.

decreasing functionIf a function decreases across the entire domain, then the function is called a decreasing function.

Example

The function shown is a decreasing function.

8

6

4

2

0

22

24

26

28

21020 4 6 8 10 12 14 16 18

x

y

degree of a polynomialThe greatest exponent in a polynomial determines the degree of the polynomial.

Example

The polynomial 2x3 1 5x2 2 6x 1 1 has a degree of 3.

degree of a termThe degree of a term in a polynomial is the exponent of the term.

In the polynomial 5x2 2 6x 1 9, the degree of the term 6x is 1.


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y extract the square rootTo extract a square root, solve an equation of the form a2 5 b for a.

extrapolationTo make predictions for values of x that are outside of the data set is called extrapolation.

F

factor an expressionTo factor an expression means to use the Distributive Property in reverse to rewrite the expression as a product of factors.

Example

The expression 2x 1 4 can be factored as 2(x 1 2).

factored formA quadratic function written in factored form is in the form f(x) 5 a(x 2 r1)(x 2 r2), where a fi 0.

Example

The function h(x) 5 x2 2 8x 1 12 written in factored form is (x 2 6)(x 2 2).

E

equivalent compound inequalitiesA compound inequality that is the equivalent of an absolute value inequality.

Examples

Absolute Value Inequality

Equivalent Compound Inequality

|ax 1 b| , c 2c , ax 1 b , c

|ax 1 b| # c 2c # ax 1 b # c

|ax 1 b| . c ax 1 b , 2c or ax 1 b . c

|ax 1 b| $ c ax 1 b # 2c or ax 1 b $ c

explicit formulaAn explicit formula of a sequence is a formula for calculating the value of each term of a sequence using the term’s position in the sequence.

Example

The sequence 1, 3, 5, 7, 9, . . . can be described by the rule an 5 2n 2 1 where n is the position of the term. The fourth term of the sequence a4 is 2(4) 2 1, or 7.

domainThe domain is the set of input values in a relation.

Example

The domain of the function y 5 2x is the set of all real numbers.

dot plotA dot plot is a graph that shows how discrete data are graphed using a number line.

Example

1

X X X XXX

XX

X

XX

XX

X X X XX

XX

2 3 4 5 6 7 8 9 10 11 12Sugar Amount in One Serving (grams)

Sugar in Breakfast Cereals

13 14 15 16 17 18 19 20 21 22

exponential functionsThe family of exponential functions includes functions of the form f(x) 5 a • bx, where a and b are real numbers, and b is greater than 0 but is not equal to 1.

Example

The function f(x) 5 2x is an exponential function.

exponentiationExponentiation means to raise a quantity to a power.


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Glossary G-7

Glossary

finite sequenceIf a sequence terminates, it is called a finite sequence.

Example

The sequence 22, 26, 30 is a finite sequence.

first differencesFirst differences are the values determined by subtracting consecutive y-values in a table when the x-values are consecutive integers.

Example

First Differences

1800 2 0 5 1800

3600 2 1800 5 1800

5400 2 3600 5 1800

Time (minutes) Height (feet)

0 0

1 1800

2 3600

3 5400

1 2 0 5 1

2 2 1 5 1

3 2 2 5 1

~~

~

five number summaryThe five number summary consists of the minimum value, the first quartile (Q1), the median, the third quartile (Q3), and the maximum value.

Example

Minimum Maximumleast valuein data set

greatest valuein the data set

Q1median of thelower half of the data set

median of theupper half of the data set

Q3

Medianmiddle value

of the data set


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function notationFunction notation is a way of representing functions algebraically.

Example

In the function f(x) 5 0.75x, f is the name of the function, x represents the domain, and f(x) represents the range.

G

geometric sequenceA geometric sequence is a sequence of numbers in which the ratio between any two consecutive terms is a constant.

Example

The sequence 2, 4, 8, 16 is a geometric sequence with a common ratio of 2.

greatest integer function (floor function)The greatest integer function, also known as a floor function is defined as the greatest integer less than or equal to x.

frequency marginal distributionA frequency marginal distribution displays the total of the frequencies of the rows or columns of a frequency distribution.

frequencyThe height of each bar in a histogram indicates the frequency, which is the number of data values included in any given bin.

functionA function is a relation between a given set of elements, such that for each element in the domain there exists exactly one element in the range.

Example

The equation y 5 2x is a function. Every value of x has exactly one corresponding y-value.

function familyA function family is a group of functions that share certain characteristics.

Examples

Linear functions and exponential functions are examples of function families.

frequency distributionA frequency distribution displays the frequencies for categorical data in a two-way table.

Example

Favorite Meals of Students

Burgers Chicken

Nuggets

Pizza

Salad Bar

Total

Gra

de

Leve

l 9th grade 4 1 3 5 13

10th grade 3 7 3 4 17

Total 7 8 6 9 30


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Glossary

horizontal asymptoteA horizontal asymptote is a horizontal line that a function gets closer and closer to, but never intersects.

Example

The graph shows a horizontal asymptote at y 5 21.

24 23 22 21

2

1

2110 2 3 4

x

22

23

24

y

4

3

horizontal translationA horizontal translation of a graph is a shift of the entire graph left or right. A horizontal translation affects the x-coordinate of each point on the graph.

Examples

The graphs of t(x 1 3) and t(x 2 1) are horizontal translations of the graph of t(x).

43

1

2

3

4

2121

2

t(x – 1)t9(x + 3) t(x)

122

22

23

23

24

24

x0

y

H

half-planeThe graph of a linear inequality is a half-plane, or half of a coordinate plane.

Example

The shaded portion of the graph is a half-plane.

18

16

14

12

10

8

6

4

2

0 1 2 3 4 5 6 7 8 9x

y

histogramA histogram is a graphical way to display quantitative data using vertical bars.

Example

Number of Hours

Hours Spent Playing Video Games on the Weekends

Num

ber

of S

tud

ents

5 15 20 25100

1

2

3

4

5

6

7

8

9

10


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independent quantityThe quantity that the dependent quantity depends upon is called the independent quantity.

Example

In the relationship between driving time and distance traveled, driving time is the independent quantity, because it does not depend on any other quantity

indexThe index is the position of a term (its term number) in a sequence. Also, that sits outside the radical is called the index of the radical.

Example

In the radical expression 4  16 , the number 4 is the index.

infinite sequenceIf a sequence continues on forever, it is called an infinite sequence.

Example

The sequence 22, 26, 30, 34 . . . is an infinite sequence.

integersThe set of integers consists of the set of whole numbers and their opposites.

Example

The numbers 212, 0, and 30 are integers.

interpolationUsing a linear regression to make predictions within the data set is called interpolation.

interquartile range (IQR)The interquartile range, IQR, measures how far the data are spread out from the median.

Example

In the data set 13, 17, 23, 24, 25, 29, 31, 45, 46, 53, 60, the median, 29, divides the data into two halves. The first quartile, 23, is the median of the lower half of the data. The third quartile, 46, is the median of the upper half of the data. The interquartile range is 46 2 23, or 23.

I

imaginary numbersThe set of imaginary numbers is the set of all numbers written in the form a 1 bi, where a and b are real numbers and b is not equal to 0.

imaginary part of a complex numberIn a complex number of the form a 1 bi, the term bi is called the imaginary part of a complex number.

imaginary roots/imaginary zerosImaginary roots are imaginary solutions to equations. Quadratic functions that do not cross the x-axis have imaginary zeros.

inconsistent systemsSystems with no solution are called inconsistent systems.

increasing functionIf a function increases across the entire domain, then the function is called an increasing function.

Example

The function shown is an increasing function.

8

6

4

2

0

22

24

26

28

21020 4 6 8 10 12 14 16 18

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Glossary

J

joint frequencyAny frequency recorded within the body of a two-way frequency table is known as a joint frequency.

L

leading coefficientThe leading coefficient of a function is the numerical coefficient of the term with the greatest power.

Example

In the function h(x) 5 27x2 1 x 1 25, the value 27 is the leading coefficient.

least integer function (ceiling function)The least integer function, also known as the ceiling function, is defined as the least integer greater than or equal to x.

least squares regression lineA least squares regression line is the line of best fit that minimizes the squares of the distances of the points from the line.

Example

The line shown is a least squares regression line for these data.

180

160

140

120

100

80

Trin

kets

Pro

duc

ed(tr

inke

ts)

60

40

20

010 2 3 4 5 6

Time (hours)

Production of Trinkets

7 8 9x

y

intersection pointIf you have intersecting graphs, a solution is the ordered pair that satisfies both functions at the same time, or the intersection point of the graphs.

Example

8100

7200

6300

5400

4500

3600

2700

1800

900

0 1 2 3 4 5 6 7 8 9Time (minutes)

Hei

ght

(feet

)

x

y h(t) 5 1800t

y 5 7200

(t, h(t))

(4, 7200)

intersection point

intervalAn interval is defined as the set of real numbers between two given numbers.

inverse functionAn inverse function takes the output value, performs some operation(s) on this value, and arrives back at the original function’s input value.

Example

The inverse of the function y 5 2x is the function x 5 2y, or y 5 x __

2 .

inverse operation“Undoing,” working backwards, or retracing steps to return to an original value or position is referred to as using the inverse operation.

Example

The operations of addition and subtraction are inverse operations.

irrational numbersThe set of irrational numbers consists of all numbers that cannot be written as a __

b where a and b are integers.

Example

The number is an irrational number.


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linear functionsThe family of linear functions includes functions of the form f(x) 5 mx 1 b, where m and b are real numbers.

Example

The function f(x) 5 3x 1 2 is a linear function.

linear piecewise functionsLinear piecewise functions include functions that have equation changes for different parts, or pieces, of the domain.

Example

The function f(x) is a linear piecewise function.

2

x 1 5 x # 22f(x) 5 22x 1 1 22 , x # 2 2x 2 9 x . 2

linear programmingLinear programming is a branch of mathematics that determines the maximum and minimum value of linear expressions on a region produced by a system of linear inequalities.

linear regressionA linear regression models the relationship between two variables in a data set by producing a line of best fit.

linear regression equationThe equation that describes the line of best fit is called the linear regression equation.

Example

The linear regression equation for these data is y 5 25x.

180

160

140

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kets

Pro

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ts)

60

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010 2 3 4 5 6

Time (hours)


7 8 9x

y

linear absolute valueThe absolute value of a number is its distance from zero on the number line.

Examples

|5| 5 5 because 5 is 5 units from 0 on the number line. |23| 5 3 because 23 is 3 units from 0 on the number line.

linear absolute value equationAn equation in the form |x 1 a| 5 c is a linear absolute value equation.

Example

The equation |x 2 1| 5 6 is a linear absolute value equation.

linear absolute value functionsThe family of linear absolute value functions includes functions of the form f(x) 5 a|x 1 b| 1 c, where a, b, and c are real numbers, and a is not equal to 0.

Example

The function f(x) 5 |x 2 3| 2 2 is a linear absolute value function.

linear absolute value inequalityAn inequality in the form |x 1 a| , c is a linear absolute value inequality.

Example

The inequality |w 2 145.045| # 3.295 is a linear absolute value inequality.

linear combinations methodThe linear combinations method is a process used to solve a system of equations by adding two equations together, resulting in an equation with one variable.

Example

Solve the following system of equations by using the linear combinations method:

  6x 2 5y 5 3

2x 1 2y 5 12  

First, multiply the second equation by 23. Then, add the equations and solve for the remaining variable. Finally, substitute y 5 3 into the first equation and solve for x. The solution of the system is (3, 3).


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Glossary

M

measure of central tendencyA measure of central tendency is the numerical values used to describe the overall clustering of data in a set.

Examples

The mean, median, and mode are the most common measures of central tendency.

monomialPolynomials with only one term are monomials.

Example

The expressions 5x, 7, 22xy, and 13x3 are monomials.

N

natural numbersThe set of natural numbers consists of the numbers that you use to count objects.

Example

The numbers 1, 2, 3, 4, . . . are natural numbers.

necessary conditionA correlation is a necessary condition for causation, meaning that for one variable to cause another, they must be correlated.

negative square rootA square root that is negative.

line of best fitA line of best fit is the line that best approximates the linear relationship between two variables in a data set.

Example

The line shown is a line of best fit for these data.

180

160

140

120

100

80

Trin

kets

Pro

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ed (t

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ts)

60

40

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010 2 3 4 5 6

Time (hours)


7 8 9x

y

line of reflectionA line of reflection is the line that the graph is reflected about.

Example

The triangle on the left was reflected over the line x 5 23 to create the triangle on the right.

literal equationLiteral equations are equations in which the variables represent specific measures.

Examples

The equations I 5 Prt and A 5 lw are literal equations.

lower fenceThe value of Q1 2 (IQR • 1.5) is known as the lower fence.


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normal distributionA normal distribution is a collection of many data points that form a bell-shaped curve.

Example

oppositesTwo numbers that are an equal distance, but are in different directions, from zero on the number line are called opposites.

Example

The opposite of 23 is 3. Both numbers are 3 units from 0 on the number line.

outlierAn outlier is a data value that is significantly greater or lesser than other data values in a data set.

Example

In the data set 1, 1, 3, 3, 4, 4, 5, 1000, the outlier is 1000.

40 60 80 100 120X X

nth rootIf n represents a positive number, then a number b is the nth root of a if bn 5 a.

Example

The number 2 is the 4th root of 16, because 24 5 16.

the number iThe number i is a number such that i2 5 21.

O

one-to-one functionA function is a one-to-one function if both the function and its inverse are functions.

Example

The equation y 5 x3 is a one-to-one function because its inverse, 3  x 5 y, is a function. The equation y 5 x2 is not a one-to-one function because its inverse, √

__ x 5 y,

is not a function.

open intervalAn open interval (a, b) describes the set of all numbers between a and b, but not including a or b.


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Glossary

Q

Quadratic FormulaThe Quadratic Formula is x 5 2b  b2 2 4ac _____________

2a .

quadratic functionsThe family of quadratic functions includes functions of the form f(x) 5 ax2 1 bx 1 c, where a, b, and c are real numbers, and a is not equal to 0.

Examples

The equations y 5 x2 1 2x 1 5 and y 5 24x2 2 7x 1 1 are quadratic functions.

quadratic regressionThe quadratic regression is a mathematical method to determine the equation of a “parabola of best fit” for a data set.

Example

The graph of the quadratic regression for these data is shown.

357

356

355

354

353

10 20 30 40 50 60 70 80 90Temperature of Can ( ̊F)

Sod

a V

olum

e (m

L)

Temperature and Volume

x

y

R

radical expressionA radical expression is an expression that involves a radical symbol ( √

__ ).

P

parabolaThe shape that a quadratic function forms when graphed is called a parabola.

Example

28 26 24 22

4

2

222

(4, 2)

0 4 6 8x

24

26

28

y

y 5 212 x2 1 4x 2 6

8

6

perfect square trinomialA perfect square trinomial is an expression in the form a2 1 2ab 1 b2 or in the form a2 2 2ab 1 b2.

polynomialA polynomial is a mathematical expression involving the sum of powers in one or more variables multiplied by coefficients.

Example

The expression 3x3 1 5x 2 6x 1 1 is a polynomial.

principal square rootA positive square root of a number.

pure imaginary numberA pure imaginary number is a number of the form bi, where b is not equal to 0.


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reflectionA reflection of a graph is a mirror image of the graph about a line of reflection.

Example

The triangle on the right is a reflection of the triangle on the left.

2

4

6

8

22

Line ofRe�ection

224224

24

26

26

28

28

6210x

0

y

relationA relation is the mapping between a set of input values called the domain and a set of output values called the range.

Example

The set of points {(0, 1), (1, 8), (2, 5), (3, 7)} is a relation.

relative frequency conditional distributionA relative frequency conditional distribution is the percent or proportion of occurrences of a category given the specific value of another category.

relative frequency distributionRepresenting the relative frequencies for joint data displayed in a two-way table is called a relative frequency distribution. The relative frequency distribution provides the ratio of occurrences in each category to the total number of occurrences.

radicandThe value that is inside a radical is called the radicand.

Example

In the radical expression √___

25 , the number 25 is the radicand.

rangeThe range is the set of output values in a relation.

Example

The range of the function y 5 x2 is the set of all numbers greater than or equal to zero.

rational exponentA rational exponent is an exponent that is a rational number written as a fraction.

Example

In the expression x 2 __ 3 , the value 2 __

3 is a rational exponent.

rational numbersThe set of rational numbers consists of all numbers that can be written as a __

b where a and b are integers, but b is

not equal to 0.

Example

The number 0.5 is a rational number because it can be written as the fraction 1 __

2 .

real numbersThe set of real numbers consists of the set of rational numbers and the set of irrational numbers.

Examples

The numbers 23, 11.4, 1 __ 2 , and √

__ 5 are real numbers.

real part of a complex numberIn a complex number of the form a 1 bi, the term a is called the real part of a complex number.

recursive formulaA recursive formula expresses each new term of a sequence based on the preceding term in the sequence.

Example

The formula an 5 an 2 1 1 2 is a recursive formula. Each successive term is calculated by adding 2 to the previous term. If a1 5 1 then a2 5 1 1 2 5 3.


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Glossary

restrict the domainTo restrict the domain of a function means to define a new domain for the function that is a subset of the original domain.

rootsThe roots of a quadratic equation indicate where the graph of the equation crosses the x-axis.

Example

The roots of the quadratic equation x2 2 4x 5 23 are x 5 3 and x 5 1.

S

second differencesSecond differences are the differences between consecutive values of the first differences.

Example

x y

23 25

22 0

21 3

0 4

1 3

2 0

3 25

First Differences

5

3

1

21

23

25

Second Differences

22

22

22

22

22

relative frequency marginal distributionDisplaying the relative frequencies for the rows or columns in a two-way table is called a relative frequency marginal distribution. The relative frequency marginal distribution provides the ratio of total occurrences for each category to the total number of occurrences.

residualA residual is the distance between an observed data value and its predicted value using a regression equation.

residual plotA residual plot is a scatter plot of the independent variable on the x-axis and the residuals on the y-axis.

Example

The graph at bottom shows a residual plot of the braking distance data.

x

y

40 60 8030 50 70 90

Bra

king

Dis

tanc

e (fe

et)

Speed (mph)

Braking Distance

20100

100

50

150

200

250

300

350

400

450

x

y

40 60 80 90

5

10

Res

idua

l Val

ue

Speed (mph)

15

20

2520 30 50 7010

0

210

215

220

Residual Plot


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skewed left distributionIn a skewed left distribution of data, the peak of the data is to the right side of the graph. There are only a few data points to the left side of the graph.

Example

These data show a skewed left distribution.

Skewed left

xxx

xxx

xx

xxxxxx

xx

xxxxxxxxx

skewed right distributionIn a skewed right distribution of data, the peak of the data is to the left side of the graph. There are only a few data points to the right side of the graph.

Example

These data show a skewed right distribution.

Skewed right

xxx

xxx

xxx

x x x xxx

xx

x x x x x x x x

slope-intercept formThe slope-intercept form of a linear equation is y 5 mx 1 b where b is the y-intercept and m is the slope.

Example

The linear equation y 5 2x 1 1 is written in slope-intercept form. The slope of the line is 2 and the y-intercept is 1.

solutionThe solution of a linear equation is any value that makes the open sentence true.

Example

The solution of the equation 3x 1 4 5 25 is 7 because 7 makes the equation true: 3(7) 1 4 5 25, or 25 5 25.

sequenceA sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects.

Example

The numbers 1, 1, 2, 3, 5, 8, 13 form a sequence.

side-by-side stem-and-leaf plotA side-by-side stem-and-leaf plot allows a comparison of two data sets. The two data sets share the same stem, but have leaves to the left and right of the stem.

Example

Difference in Departure Times (minutes)

My Air Airlines Fly High Airlines

5 0 0 7 8

9 5 1 1 4 5 6

6 0 0 2 4 7 9

4 3 3 3 0 2

0 4 5 9

24 5 24 minutes

significant digitsSignificant digits are digits that carry meaning contributing to a number’s precision.

simple interestIn a simple interest account, the interest earned at the end of each year is a percent of the original deposited amount (also known as the original principal).

Example

Tonya deposits $200 in a 3-year certificate of deposit that earns 4% interest. The amount of interest that Tonya earns can be found using the simple interest formula.

I 5 (200)(0.04)(3) I 5 24 Tonya earns $24 in interest.


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Glossary

square rootA number b is a square root of a if b2 5 a.

standard deviationStandard deviation is a measure of how spread out the data are from the mean.

standard formThe standard form of a linear equation is Ax 1 By 5 C where A, B, and C are constants and A and B are not both zero.

Example

The linear equation 2x 1 3y 5 5 is written in standard form.

standard form (general form) of a quadratic functionA quadratic function written in the form f(x) 5 ax2 1 bx 1 c, where a fi 0, is in standard form, or general form.

Example

The function f(x) 5 25x2 2 10x 1 1 is written in standard form.

statisticStatistics are numerical characteristics of data.

stem-and-leaf plotA stem-and-leaf plot is a graphical method used to represent ordered numerical data.

Example

A stem-and-leaf plot can be drawn to represent test scores.

55, 62, 73, 75, 76, 79, 80, 83, 86, 87, 87, 88, 88, 89, 89, 89

The tens’ place represents the stem and the ones’ place represents the leaves.

Stems Leaves

1 2 3 4 5 5 6 2 7 3 5 6 9 8 0 3 6 7 7 8 8 9 9 9

Key: 7 | 3 5 73

solution of a compound inequalityThe solution of a compound inequality is the part or parts of the solutions that satisfy both of the inequalities.

Example

The number line shows the solution of the compound inequality x , 22 or x . 1.

–5 –4 –3 –2 –1 0 1 2 3 4 5

x < –2 or x > 1

solution of a system of linear inequalitiesThe solution of a system of linear inequalities is the intersection of the solutions to each inequality. Every point in the intersection region satisfies the solution.

Example

The solution of this system of linear inequalities . . .

  200a 1 100c # 800 75(a 2 1) 1 50c $ 150

 

. . . is shown by the shaded region, which represents the intersection of the solutions to each inequality.

Adult Rafters

Chi

ld R

afte

rs

86

2

4

6

8

2222

4224

24

26

26

28

28

x0

y

solve an inequalityTo solve an inequality means to determine the values of the variable that make the inequality true.

Example

The inequality x 1 5 . 6 can be solved by subtracting 5 from each side of the inequality. The solution is x . 1. Any number greater than 1 will make the inequality x 1 5 . 6 true.


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symmetric distributionIn a symmetric distribution of data, the left and right halves of the graph are nearly mirror images of each other. There is often a “peak” in the middle of the graph.

Example

These data show a symmetric distribution.

Symmetric

x xx x x x xx x x x

x xx x

xx

xx x

x

system of linear equationsWhen two or more equations define a relationship between quantities, they form a system of linear equations.

Example

The equations y 5 3x 1 7 and y 5 24x are a system of equations.

T

termWithin a polynomial, each product is a term.

Example

The polynomial 2x 1 3y 1 5 has three terms: 2x, 3y, and 5.

term of a sequenceA term of a sequence is an individual number, figure, or letter in the sequence.

Example

In the sequence 2, 4, 6, 8, 10, the first term is 2, the second term is 4, and the third term is 6.

step functionA step function is a piecewise function whose pieces are disjoint constant functions.

Example

x

3

2

1

0654

Distance Traveled (miles)10 2 3 87 9

4

Taxi

Far

e (d

olla

rs)

8

7

6

5

9

y

substitution methodThe substitution method is a process of solving a system of equations by substituting a variable in one equation with an equivalent expression.

Example

Solve the following system of equations by using the substitution method:

  x 2 3y 5 4

2x 1 5y 5 214  

First, solve the first equation for x. Then, substitute in the second equation. Next, substitute y 5 22 into the equation x 2 3y 5 4. The solution of the system is (22, 22).

sufficient conditionA correlation is not a sufficient condition for causation, meaning that a correlation between two variables is not enough to establish that one variables causes another.

sum of two cubesThe sum of two cubes is an expression in the form a3 1 b3 that can be factored as (a 1 b)(a2 2 ab 1 b2).


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Glossary G-21

Glossary

U

upper fenceThe value of Q3 1 (IQR • 1.5) is known as the upper fence.

V

Venn diagramA Venn diagram uses circles to show how elements among sets of numbers or objects are related.

Example

Whole numbers 1–10

7

8

91

2

3

45

10

Factorsof 30

Factorsof 18

6

transformationA transformation is the mapping, or movement, of all the points of a figure in a plane according to a common operation.

Example

trinomialPolynomials with exactly three terms are trinomials.

Example

The polynomial 5x2 2 6x 1 9 is a trinomial.

two-way frequency tableA two-way frequency table displays categorical data by representing the number of occurrences that fall into each group for two variables.

Example

Favorite Meals of Students

Burgers Chicken Nuggets

Pizza Salad Bar

Gra

de

Leve

l 9th grade    4

1

  3 5

10th grade   3 7

  3

   4

re�ection in a line rotation about a point


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G-22 Glossary

Glo

ssar

y

Vertical Line TestThe Vertical Line Test is a visual method used to determine whether a relation represented as a graph is a function.

Examples

The equation y 5 3x2 is a function, because the graph of the function passes the Vertical Line Test.

24 23 22 21

10

5

2510 2 3 4

x

210

y

30

25

20

15

The equation x2 1 y2 5 9 is not a function, because the graph of the function fails the Vertical Line Test.

24 23 22 21

2

1

2110 2 3 4

x

22

23

24

y

4

3

vertexThe vertex of a parabola is the lowest or highest point on the curve.

Example

The vertex of the graph of y 5 2 __ 3 x2 2 4 __

5 x 2 10 ___

3 is the

point (1, 24), the absolute minimum of the parabola.

28 26 24 22

4

2

222

(1, 24)

0 4 6 8x

24

26

28

y

23y 5 x2 2

43

103x 2

8

6

vertex formA quadratic function written in vertex form is in the form f(x) 5 a(x 2 h)2 1 k, where a fi 0.

Example

The quadratic equation y 5 2(x 2 5)2 1 10 is written in vertex form. The vertex of the graph is the point (5, 10).

vertical dilationA vertical dilation of a function is a transformation in which the y-coordinate of every point on the graph of the function is multiplied by a common factor.

Example

The coordinate notation (x, y) → (x, ay), where a is the dilation factor, indicates a vertical dilation.


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Glossary G-23

Glossary

W

whole numbersThe set of whole numbers consists of the set of natural numbers and the number 0.

Example

The numbers 0, 1, 2, 3, . . . are whole numbers.

Z

Zero Product PropertyThe Zero Product Property states that if the product of two or more factors is equal to zero, then at least one factor must be equal to zero. This is also called the Converse of Multiplication Property of Zero.

zerosThe x-intercepts of a graph of a quadratic function are also called the zeros of the quadratic function.

Example

The zeros of the quadratic function f(x) 5 22x2 1 4x are (0, 0) and (2, 0).

vertical motion modelA vertical motion model is a quadratic equation that models the height of an object at a given time. The equation is of the form g(t) 5 216t2 1 v0t 1 h0, where g(t) represents the height of the object in feet, t represents the time in seconds that the object has been moving, v0 represents the initial velocity (speed) of the object in feet per second, and h0 represents the initial height of the object in feet.

Example

A rock is thrown in the air at a velocity of 10 feet per second from a cliff that is 100 feet high. The height of the rock is modeled by the equation y 5 216t2 1 10t 1 100.

vertical translationA vertical translation is a type of transformation that shifts the entire graph up or down. A vertical translation affects the y-coordinate of each point on the graph.

Example

The graphs of z(x) 1 3 and z(x) 2 4 are vertical translations of the graph of z(x).

24 23 22 2121

22

23

10 2 3 4x

24

y

4

3

2

1

z(x) + 3

z(x) – 4

z(x)


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Index I-1

Arithmetic sequences, 229–230, 288common difference of, 229–230, 288

as slope of linear function, 291definition of, 229explicit formulas for, 238–239, 277,

289in function notation, 277, 298graphs of, 290recursive (NOW NEXT) formulas for,

245–246, 289Association, types of, 534

See also CorrelationAssociative Property, 85

of Addition, 846, 871functions and, 275–286of Multiplication, 871as tables, 252–255

Asymptotes, 309, 357Axes of graphs, labeling, 74Axis of symmetry, 656–660, 694, 867

completing the square to identify, 777–778

symmetric points to determine, 694

BBar graph, 610, 612Bar notation, 843Base

common, 352of exponential function, 291

Basic function, 314, 358, 359, 361Best fit

line of, 167, 203parabola of, 808regression equation of, 941, 949, 952,

954, 975Bin, 463Binomials, 705

as difference of two squares, 752, 783, 784

multiplying, 718–727algebra tiles for, 718–720Distributive Property for, 725–726FOIL method of, 727for modeling, 718–722, 780–781multiplication table for, 723–724

Box-and-whisker plots, 460, 473, 483, 514

outliers on, 483Break-even point, 368–369

CCalculator-based ranger (CBR), 803–812Categorical (qualitative) data, 581, 609Categorical variables, data sets for two,

577–614drawing conclusions from, 601–608,

613–614

See also Frequency distributionsCatenary function, 617Causation

correlation vs., 563–568, 575definition of, 565proving, 565relationships mistaken for, 566

Ceiling function (least integer function), 893, 894, 924

Central tendency, measures of, 469–478definition of, 470mean, 470, 472, 476, 503, 510, 516median, 470, 476, 503, 510, 516

Change, rate ofnegative, 155for simple and compound interest,

356unit, 76–77, 150

Closed interval, 641, 692Closed set of numbers (closure), 838,

852, 870Coefficient

of determination, 809–810, 831leading, 632

negative, 647, 693in polynomial, 704See also Correlation coefficient

(r-value)Collinear points, 527Common base, 352Common difference, 229–230, 288

as slope of linear function, 291Common ratio, 231, 288

as base of exponential function, 291Common response, 566Commutative Property, 85

of Addition, 846, 871of Multiplication, 846, 871

Completing the square, 771–778axis of symmetry and vertex of

quadratic functions identified by, 777–778

definition of, 774roots of quadratic equation by, 776,

786Complex numbers

imaginary part of, 856notation, 856real part of, 856set of, 856, 863, 872

properties of, 872Complex roots (zeros) of quadratic

equations, 863, 872–873Composition of functions, 902–903, 926Compound inequalities, 42, 111–122

in compact form, 114, 120, 157conjunction, 117, 156

Ind

ex

Index

SymbolsC (complex numbers), 856∀ (for all), 846I (imaginary numbers), 856Z (integers), 856J (irrational numbers), 856,, #, $, . (inequality), 413, 414, 422´ (is an element of/are elements of ), 846x (mean), 470N (natural numbers), 856p (pi), 469√ (radical), 764Q (rational numbers), 856R (real numbers), 846, 856s (standard deviation), 492S (summation), 470W (whole numbers), 856

AAbsolute minimum and absolute

maximum of functions, 40–41, 67quadratic function, 621, 687–688

Absolute value, definition of, 124, 127Absolute value equations, linear,

127–129isolating absolute value parts of, 129solutions of, 125, 127–129, 157writing and evaluating, 158

Absolute value functions, linear, 42, 50, 56, 68

constructed from linear piecewise function, 883, 922

graph of, 133–134Absolute value inequalities, linear,

130–134forms of, 134graph of, 158–159on a number line, 159solving, 134, 158–159writing, 130–131

AdditionAssociative Property of, 846, 871Commutative Property of, 846, 871of negative numbers, 229of polynomials, 703–716, 780

Addition Property of Equality, 85Additive identity, 847, 871Additive inverse, 839, 847, 871Algebraic expression, factoring, 646, 693Algebraic solutions

of linear functions, 78–80of linear inequalities, 106–107

Algebra tiles, to multiply binomials, 718–720

Area model, to factor trinomial, 734–735Argument of function, 319Arithmetic mean, 470, 472, 476, 503,

510, 516

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I-2 Index

Ind

ex

Division by zero, 840Division Property of Equality, 85Domain

of quadratic function, 690of relation, 32restricting, 916–917, 928

Dot plots, 457, 513

EEquations

analyzing, 78–80relationship between function

notation and, 36, 66for solving linear situations,

comparing tables, and graphs to, 153

with two variables, 205, 206See also Linear equations; Literal

equationsEquivalent compound inequalities, 134Experiments, 564Explicit formulas

for arithmetic sequences, 238–239, 277, 289

in function notation, 277, 282for geometric sequences, 242, 282,

289Exponential functions, 40, 48, 56, 68,

293–364base of, 291exponential growth, 348–351graphs of, 305–312, 357, 364

asymptotes of, 309, 357reflections of, 327–331, 361–362transformations of, 316–321,

358–362increasing and decreasing, 356interest

compound, 300–302, 306, 355–356rate of change for, 356simple, 296–299, 302, 355–356

inverses, 353–354linear functions compared to,

295–304modeling using, 931–937, 971multiple representations of, 310–312nth roots in radical form, 341, 363population problems, 356, 357radical expressions, 363radicals as powers and powers as

radicals, 363rational exponents, 337–346, 363

definition of, 343solving, 347–354, 364

using properties of exponents and common bases, 352

writing, 364Exponential regression equation, 933, 971Exponentiation, 852Exponents

negative, 338–339, 363rational, 337–346solving exponential functions using

properties of, 352Expressions, writing, 149Extracting the square root, 763, 769, 785Extrapolation, 526, 569–570

selecting best data, 809variables in vs. groups in, 581

Data sets for one variable, 453–520analyzing and interpreting, 505–512graphically representing, 455–468

box-and-whisker plots, 460, 473, 483, 514

data distribution, 458–459, 513, 516

dot plots, 457, 513histograms, 463, 515side-by-side stem-and-leaf plot,

508, 520interquartile range (IQR), 481, 492,

503, 510, 517measures of central tendency,

469–478definition of, 470mean, 470, 472, 476, 503, 510, 516median, 470, 476, 503, 510, 516

outliers, 482–483, 492, 502, 517standard deviation, 489–504, 510,

536calculating and interpreting, 518definition of, 492formula for, 492–493, 518graphing calculator to determine,

495graph of, 497–498, 501, 519

Data sets for two categorical variables, 577–614

drawing conclusions from, 601–608, 613–614

See also Frequency distributionsDecimal

percent converted to, 296repeating, 843–844, 870

Decreasing functions, 38–39, 66Degree of polynomial, 705Degree of term, 705Dependency, 4–8Dependent quantities, 4–8, 36, 63

identifying, 149Difference of two cubes, 756, 784Difference of two squares, 752, 783Dilation factor, 682, 699Dilations of quadratic functions,

681–683, 699Discontinuous piecewise functions, 923

See also Step functionsDiscrete data, 457, 513Discrete graph, 32, 64, 65Discriminant, 797–801, 830

negative, 872Disjunctions, 117, 156Distributive Property

to combine linear equations, 209of Division over Addition, 871of Division over Subtraction, 871of Multiplication over Addition, 847,

871of Multiplication over Subtraction, 871for multiplying polynomials, 725–726,

780–781in reverse, factoring as, 646to rewrite quadratic expression, 619to write linear equations, 208

definition of, 113disjunction, 117, 156equivalent, 134examples of, 114“or” or “and” in, 113–114, 116, 118,

119, 120, 156, 157solution of, 117–120

more than one, 115–117on number line, 117–118, 156

table of, 112–113writing, 156

Compound interest, 300–302, 355–356formula for, 302, 306rate of change for, 356

Conclusions, 564Confounding variable, 566Conjunctions, 117, 156Consistent system of linear equations,

381Constant functions, 38–39, 66Constraints, 420, 448Continuous data, 463, 515Continuous graph, 32, 64, 65Converse of Multiplication Property of

Zero (Zero Product Property), 744, 782

Coordinate notation, 315horizontal translation using, 320, 360reflections using, 329–330, 362vertical translation using, 315, 358

Coordinate planeinequalities represented on, 155points on, 9, 64

Coordinates of intersection point of two graphs, 369

Coordinate system, 251Correlation, 533–540, 571

causation vs., 563–568, 575definition of, 535

Correlation coefficient (r-value), 169, 204, 535, 542, 543, 574, 933

analyzing correlation using, 571coefficient of determination and, 809determining and interpreting, 571formula for, 536, 571

Counterexample, 839Cube root, 341, 363Cubes

difference of two, 756, 784sum of two, 757, 784

DData

categorical, 581, 609continuous, 463, 515discrete, 457, 513interval of, 515raw, 613representing, 584–586

Data distribution, 458–459normal, 498, 519skewed, 458, 459, 476, 513, 516

Data setsquadratic regression of, 808–812, 830

coefficient of determination, 809–810, 831

parabola of best fit, 808

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Index I-3

Ind

ex

value feature, 96, 449zeros of quadratic function on, 640

Graphsanalyzing and sorting, 17–34bar, 610, 612continuous, 32, 64, 65coordinates of intersection point of

two, 369of data sets for one variable, 455–468

box-and-whisker plots, 460, 473, 483, 514

data distribution, 458–459, 513, 516

dot plots, 457, 513histograms, 463, 515side-by-side stem-and-leaf plot,

508, 520discrete, 32, 64, 65of exponential functions, 305–312,

357, 364asymptotes of, 309, 357reflections of, 327–331, 361–362transformations of, 316–321,

358–362of frequency distribution, 584–586,

610of functions, 56–62intersection point of, 82, 105key for, 585, 610labeling axes of, 74of linear absolute value function,

133–134of linear absolute value inequalities,

158–159of linear equations, 150, 178–180of linear functions, 78–80of linear inequalities, 104–105, 108,

411–418, 447as half-plane, 414, 447representing solution set in, 414–

417, 447of linear inverse functions, 900–901,

925of linear piecewise function, 887, 921of linear regression equation, 168matching scenarios and, 9–14, 64of nonlinear functions, 160of non-linear inverse functions,

914–916of normal distribution, 501of quantities, 4–8of relative frequency distribution, 612of sequences, 251–274, 290for solving linear situations,

comparing tables and equations to, 153

solving systems of quadratic equations with, 822–828

of standard deviation, 497–498, 501, 519

of step function, 891–894, 923of system of linear equations, 403,

406of systems of linear inequalities, 433,

449with more than two linear

inequalities, 450

decreasing, 38–39, 66definition of, 32families of, 40–52

distinguishing between, 54–55, 67–68, 69

graphical behaviors of, 55–62, 70graphing calculator to graph, 37increasing, 38–39, 66leading coefficient of, 632linear absolute value, 883, 922linear piecewise, 877–883, 887, 921naming, 376one-to-one, 910–913, 927as relation, 65representing linear inequalities, 103,

110restricting domain of, 916–917, 928sequences and, 275–286in two variables, 174–186, 205See also Exponential functions;

Inverse functions; Linear functions; Quadratic functions

GGeneral form. See Standard formGeometric sequences, 231, 288

common ratio of, 231, 288as base of exponential function,

291definition of, 231explicit formulas for, 242, 289

in function notation, 282graphs of, 290recursive formulas for, 246, 289

Graphic organizers, 45Graphing calculator

absolute maximum of quadratic function on, 621

box-and-whisker plot on, 473calculator-based ranger (CBR) with,

805–812to complete a table, 95data set represented in, 165exponential regression equation for

data set on, 933, 971function graphed on, 37graphs analyzed with, 96–98greatest integer function on, 894intersect feature of CALC, 97–98intersection feature, 357, 364least integer function on, 894linear regression on, 167, 203mean on, 472parentheses on, 99piecewise functions graphed on, 43to plot residuals, 558product of two binomials on, 720quadratic regression on, 808–812

coefficient of determination on, 809–810, 831

selecting best data, 809recursive formula on, 248–249scientific notation on, 42standard deviation determined using,

495step function on, 891TABLE feature of, 95–96

FFactor, dilation, 682, 699Factored form, quadratic functions in,

645–652, 693, 696, 866definition of, 647given its x-intercepts, 693identifying characteristics of parabola

given, 696Factoring, 646

polynomials, 731–742by determining greatest common

factor, 732–733, 781trinomials, 733–741, 782

quadratic equations, 743–749, 782–783

to calculate roots, 746, 782Families of functions, 40–52

distinguishing between, 54–55, 67–68, 69

graphical behaviors of, 55–62, 70Fibonacci sequence, 221Finite sequences, 220, 287First differences, 76

definition of, 630to determine unit rate of change, 150

Five number summary, 460, 514Floor function (greatest integer function),

892, 894, 924FOIL method of multiplying polynomials,

727Formulas, literal equations in, 188,

193–194Fraction, writing repeating decimal as,

843–844, 870Frequency, 463, 515

joint, 582Frequency distributions, 579–614

definition of, 582frequency marginal distribution, 583,

610graphs of, 584–586, 610interpreting, 579–588relative, 589–594

creating and analyzing, 611definition of, 590, 611graphs of, 612

relative frequency conditional distribution, 595–600

creating and analyzing, 613definition of, 598

relative frequency marginal distribution, 590, 611

two-way frequency table to analyze, 581, 609

Function notation, 36, 66arithmetic sequence in, 298explicit formula in, 277, 282solutions to linear equations using,

150Functions

absolute minimum and absolute maximum of, 40–41, 67

adding, 201argument of, 319basic, 314, 358, 359, 361composition of, 902–903, 926constant, 38–39, 66

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converting to solve for specific variable, 207

Distributive Property of Multiplication to write, 208

graphing solution for, 150, 178–180in slope-intercept form, 183, 184,

191, 207solution of, 82

on a graph using an intersection point, 151

using function notation, 150in standard form, 183–186, 207system of quadratic equation and,

822–824, 832–833table values for, 185–186unit analysis of, 181–183See also Systems of linear equations

Linear expressions, maximum and minimum values of, 439–446, 451

Linear functions, 40, 47, 56, 68, 73–101, 161–210

exponential functions compared to, 295–304

graph of, 78–80inverses of, 895–907, 924–926

algebraic method of determining, 899, 925

compositions of functions to determine, 903, 926

graphs of, 900–901, 925notation, 899, 925

parts of, 152quadratic functions compared with,

625–636, 688–689leading coefficient, 632second differences, 689

slope of, common difference as, 291solutions of

algebraic, 78–80graphing, 78–80using intersection points, 82–84

tables for, 74–77technology to complete, 95–96

transformations of, 322–324reflections, 331–332, 361, 362translations, 322–324, 358–360

units of measure associated with, 74–79

See also Linear regression equationsLinear inequalities, 101–110

algebraic solution for, 106–107function representing, 103, 110graph of, 104–105, 108, 411–418

as half-plane, 414, 447representing solution set in, 414–

417, 447reversing sign of, 107–109in two variables, 447See also Systems of linear

inequalitiesLinear piecewise functions, 42–43, 51,

68, 877–883, 921constructing absolute value function

from, 883, 922defined, 879, 921graphing, 887, 921

coordinates of, 369linear equations solved using, 150linear functions solved using, 82–84non-linear functions solved using, 160population problems solved using,

357Interval notation, 641Intervals

closed, 641, 692of data, 515defined, 641, 692half-closed or half-open, 641, 692of increase and decrease, 642–643,

692open, 641, 692

Inverse, 353–354additive, 839, 847, 871multiplicative, 847, 871

Inverse functionsdefinition of, 899interpreting, 972–973linear, 895–907, 924–926

algebraic method of determining, 899, 925

compositions of functions to determine, 903, 926

graphs of, 900–901, 925notation, 899, 925

non-linear, 909–920graph of, 914–916one-to-one, 910–913, 927of quadratics, 914, 928with table of values, 911–913

Inverse operation, 897, 924Irrational numbers, 841, 843, 869

JJoint frequency, 582

KKey for graphs, 585, 610

LLabeling axes of graphs, 74Leading coefficient, 632

negative, 647, 693Least integer function (ceiling function),

893, 894, 924Least squares regression, 523–532

See also Linear regressionLeast squares regression equation, 570,

574Least squares regression line, 528, 570Linear absolute value equation. See

Absolute value equations, linearLinear absolute value functions. See

Absolute value functions, linearLinear absolute value inequalities. See

Absolute value inequalities, linearLinear combinations method of solving

system of linear equations, 383–390, 405, 406

Linear equations, 173–186combining, 195–202

Distributive Property of Multiplication for, 209

using graphing calculator, 426, 449technology to analyze, 96–98transformations of. See

Transformationsvisual display of data in, 9, 64

Greatest common factorfactoring, from algebraic expression,

646, 693factoring polynomials by determining,

732–733, 781Greatest integer function (floor function),

892, 894, 924

HHalf-closed or half-open interval, 641,

692Half-plane, 414, 447Histograms, 463, 515Horizontal asymptote, 309Horizontal reflections, 362Horizontal translations, 319–321,

359–360, 697

IImaginary numbers, 851–857

i, 853–855notation, 856pure, 856set of, 856, 872simplifying expressions involving,

855, 872Imaginary part of complex number, 856Imaginary roots (zeros), 863Inconsistent system of linear equations,

381Increasing functions, 38–39, 66Independent quantities, 4–8, 36, 63, 75

identifying, 149Index

of radical, 341, 363of term, 238

Inequalitiesdefinition of, 413with a negative rate of change, 155representing on a coordinate plane,

155representing on a number line, 154simple, 113solving, 154writing, 154See also Compound inequalities;

Linear inequalities; Quadratic inequalities

Inequality symbols, 413, 414, 422Infinite sequences, 220, 287Infinity, negative, 42Integers, 839, 869Intercepts of equation with two

variables, 205Interest

compound, 300–302, 306, 355–356rate of change for, 356simple, 296–299, 302, 355–356

Interpolation, 525, 569–570Interquartile range (IQR), 481, 492, 503,

510, 517Intersection points, 82, 105

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Index I-5

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on graphing calculator, 99during substitution, 791

Patterns, recognizing and describing, 213–222, 287

See also SequencesPercent, calculations with, 296Perfect square trinomial, 752, 783Piecewise functions

discontinuous, 923linear, 42–43, 51, 68, 877–883, 921

constructing absolute value function from, 883, 922

defined, 879, 921graphing, 887, 921writing, 880, 921

modeling with, 963–970, 976See also Step functions

Pointsbreak-even, 368–369collinear, 527on coordinate plane, 9, 64See also Intersection points

Polynomialsadding and subtracting, 703–716, 780coefficient of, 704defined, 704degree of, 705factoring, 731–742

by determining the greatest common factor, 732–733, 781

trinomials, 733–741, 782identifying characteristics of, 779multiplying, 717–730

FOIL method, 727modeling by, 718–722, 780–781using algebra tiles, 718–720using Distributive Property for,

725–726, 780–781using multiplication table, 723–724

in one variable, 704in standard form, 709terms of, 704, 705

Population problems, 356intersecting points to solve, 357

Powers, as radicals, 363Principal (positive) square root, 763, 785Pronic number, 837Pure imaginary number, 856

QQuadratic equations

factoring, 743–749, 782–783fit of, 808–810, 831roots of, 744–749

completing the square to determine, 776, 786

complex, 863, 872–873definition of, 746factoring to calculate, 746, 782number of, 799–801, 830quadratic formula to determine,

790, 829systems of, 821–828

with linear equation, 822–824, 832–833

with two quadratic equations, 824–828, 833

using Distributive Property for, 725–726, 780–781

using multiplication table, 723–724Multiplication Property of Equality, 85Multiplication table

to calculate product of binomials, 723–724

to factor trinomials, 736–737Multiplicative identity, 847, 871Multiplicative inverse, 847, 871

NNatural numbers, 838, 869Necessary condition, 565, 575Negative exponents, 338–339, 363Negative infinity, 42Negative numbers, adding, 229Negative square root, 763, 785Non-linear functions

graphically solving using intersection points, 160

inverse, 909–920graph of, 914–916one-to-one, 910–913, 927quadratics, 914, 928with table of values, 911–913

Normal distribution, 498, 519graph of, 501

NOW NEXT formulas. See Recursive formulas

nth root, 341, 363Number i, 853–855

See also Imaginary numbersNumber laws. See Associative Property;

Commutative Property; Distributive Property

Number lineinequalities represented on, 154linear absolute value inequality on,

159opposites on, 124, 125solution of compound inequality on,

117–118, 156

OOne-to-one functions, 910–913, 927Open interval, 641, 692Opposites, 124, 125Ordered pairs, unit rate of change from,

76Outliers, 482–483, 492, 502, 517

PParabola, 621, 683–685, 687

axis of symmetry of, 656–660, 694, 867

of best fit, 808equation of, given information about

its graph, 696identifying characteristics of, 662–

667, 696meaning of word, 625vertex of, 656–660, 694

determining symmetric points using, 695

See also Quadratic functionsParentheses

writing, 880, 921Linear programming, 439–446, 451

definition of, 440Linear regression, 163–172

analyzing, 170–171correlation coefficient of, 169, 204,

535, 536, 542, 543, 571, 574definition of, 167, 203graph of, 168

using graphing calculator, 167, 203interpolation and extrapolation from,

525, 569–570to make predictions, 164–166, 203to model data, 163–172, 203

Linear regression equations, 167graphing calculator to determine,

167graph of, 168interpreting, 569

Linear regression line, correlation and, 535

Linesof best fit, 167, 203linear regression, 535of reflection, 329, 361, 362See also Number line

Literal equations, 187–194definition of, 188in formulas, 188, 193–194in slope-intercept form, 191–192in standard form, 191–192

Lower fence, 482Lucky numbers, 837

MMathematical modeling, 929–976

with binomials, 718–722, 780–781with exponential functions, 931–937,

971linear regression for, 163–172, 203with piecewise functions, 963–970,

976with quadratic functions, 944–955,

974of quadratic motion, 803–812with regression equation, 939–944,

957–962regression equations for, 939–944

Maximum of functions, absolute, 40–41, 67

Mean, 470, 472, 476, 503, 510, 516Measures of central tendency. See

Central tendency, measures ofMedian, 470, 476, 503, 510, 516Minimum of functions, absolute, 40–41,

67Modeling. See Mathematical modelingMonomials, 705Motion, quadratic. See Quadratic motion,

calculator-based ranger to modelMultiplication

Associative Property of, 871Commutative Property of, 846, 871of polynomials, 717–730

FOIL method, 727modeling by, 718–722, 780–781

using algebra tiles, 718–720

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Relative frequency conditional distribution, 595–600

creating and analyzing, 613definition of, 598

Relative frequency distribution, 589–594creating and analyzing, 611definition of, 590, 611graphs of, 612

Relative frequency marginal distribution, 590, 611

Repeating decimals, 843–844, 870Repunit number, 837Residual plots, 541–562

creating, 541–552, 572definition of, 545graphing calculator to create, 558interpreting, 546shape of, 546, 573, 574using, 553–562

Residuals, 543Response, common, 566Restricting the domain, 916–917, 928Roots

imaginary, 863of quadratic equations, 744–749

completing the square to determine, 776, 786

complex, 863, 872–873definition of, 746factoring to calculate, 746, 782number of, 799–801, 830quadratic formula to determine,

790, 829of quadratic functions, 691See also Zeros

r-value. See Correlation coefficient (r-value)

SScatter plot, 542, 574

See also Residual plotsScenarios, matching graphs to, 9–14, 64Scientific notation, on graphing

calculator, 42Second differences, 632–635, 689Sequences, 211–292

definition of, 214, 287Fibonacci, 221finite, 220, 287general rules for expressing, 238graphs of, 251–274infinite, 220, 287recognizing, 213–222terms in, 214, 287

formulas to determine, 235–250index of, 238initial, 238nth, 238term number, 238, 239

See also Arithmetic sequences; Geometric sequences

Sets, closed, 838, 852, 870See also Data sets; Data sets for

one variable; Data sets for two categorical variables

Side-by-side stem-and-leaf plot, 508, 520

quadratic regression of set of data, 808–812, 830

steps for, 805Qualitative (categorical) data, 581, 609Quantities, 1–16

dependent, 4–8, 36, 63, 149graphs of, 4–8independent, 4–8, 36, 63, 75, 149

RRadical expressions, 363, 764

approximating value of, 765rewriting, 767

Radicalsindex of, 341, 363as powers, 363

Radicand, 341, 764Range

of quadratic functions, 690of relation, 32

Rate of changenegative, 155for simple and compound interest,

356unit, 76–77, 150

Rational exponents, 337–346, 363definition of, 343

Rational numbers, 840, 869repeating decimal, 843–844, 870

Raw data, 613Real number system, 837–844, 869

numbers of, 837–844, 869closure for, 838, 852, 870

properties of, 845–849, 871simplifying expressions using,

848–849, 871Real part of complex number, 856Recursive formulas

for arithmetic sequences, 245–246, 289

for geometric sequences, 246, 289on graphing calculator, 248–249

Reflections, 327–336of exponential functions, 328–331,

361–362horizontal, 362of linear functions, 331–332, 361–362line of, 329, 361, 362of quadratic functions, 678–680, 698vertical, 361–362

Regression equationsof best fit, 941, 949, 952, 954, 975to model data, 939–944, 957–962See also Linear regression equations

Relationsdefinition of, 32domain of, 32function as, 65range of, 32

Relationshipsdependency in, 4–8between equation and function

notation, 36mistaken for causation, 566points on coordinate plane modeling

or representing, 9, 64See also Functions; Graphs

vertical motion model described by, 638–640, 654

Quadratic Formula, 789–802, 829deriving, 792discriminant in, 797–801, 830, 872solving efficiently with, 801–802using, 793–796

Quadratic functions, 42, 49, 56, 68, 615–700

absolute maximum and minimum of, 621, 687–688

axis of symmetry of, 656–660, 694, 867

completing the square to identify, 777–778

symmetric points to determine, 694domain of, 690equation of inverse of, 914, 928in factored form, 645–652, 693, 696,

866definition of, 647given its x-intercepts, 693identifying characteristics of

parabola given, 696intervals of increase and decrease of,

642–643, 692linear functions compared with, 625–

636, 688–689leading coefficient, 632second differences, 689

to model data, 944–955, 974range of, 690in standard form, 619, 687

identifying axis of symmetry and vertex of, 777–778

identifying characteristics of parabola given, 696

transformations of, 675–686dilations, 681–683, 699reflections, 678–680, 698translations, 676–678, 697writing equations given, 700

in vertex form, 661–671, 695, 696, 867

vertex of, 653–660, 694, 695completing the square to identify,

777–778defined, 656symmetric points on parabola

determined using, 695x-intercepts of graph of, 639–640,

691, 693, 783zeros (roots) of, 640, 691, 772–773,

783completing the square for, 774complex, 863, 872–873number of, 797–801, 830quadratic formula to determine,

791, 829See also Parabola

Quadratic inequalities, 813–820solution set to, 816, 832writing, 831

Quadratic motion, calculator-based ranger to model, 803–812

coefficient of determination, 809–810, 831

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Transformations, 313–326, 358–362characteristics of graphs after, 333–

336, 362definition of, 315, 329of exponential functions, 316–321,

358–362reflections, 327–331, 361–362translations, 313–326

horizontal translation, 319–321, 359–360, 697

of linear functions, 313–336reflections, 328–331, 361–362translations, 313–326, 358–360

of quadratic functions, 675–686dilations, 681–683, 699reflections, 678–680, 698translations, 676–678, 697writing equations given, 700

of systems of linear equations, 377–379

using coordinate notation, 315, 320, 329–330, 358, 360, 362

vertical translation, 314–318, 358–359, 697

Translations. See TransformationsTrial and error method to factor

trinomials, 737–741Trinomials, 705

factoring, 733–741, 782area model for, 734–735multiplication table for, 736–737trial and error method for,

737–741perfect square, 752, 783

Two-way frequency table, 581, 609

UUnit analysis, 181–183Unit rate of change, 150Upper fence, 482

VVariables

confounding, 566in data set, 581See also Categorical variables, data

sets for twoVenn diagram, 841Vertex form of quadratic functions,

661–671, 695, 696, 867identifying characteristics of parabola

given, 662–667, 696Vertex of quadratic functions, 653–660,

694defined, 656determining symmetric points on

parabola using, 695identifying, by completing the square,

777–778Vertical dilation, 682, 699Vertical Line Test, 32, 65Vertical motion model, 638–640, 654Vertical transformations

reflections, 361–362translations, 314–318, 358–359,

697

Substitution, using parentheses during, 791

Substitution method for solving systems of linear equations, 370, 404, 406

Subtraction of polynomials, 703–716, 780

Subtraction Property of Equality, 85Sufficient condition, 565, 575Sum of two cubes, 757, 784Symmetric data distribution, 458, 459,

476, 513, 516Systems of linear equations, 365–408

consistent, 381definition of, 370, 403graphing method of solving, 403, 406inconsistent, 381linear combinations method of

solving, 383–390, 405, 406to represent problem context, 406solution of, 403in standard form, 377substitution method for solving, 370,

404, 406transforming, 377–379

Systems of linear inequalities, 419–452constraints in, 420, 448graph of, 426, 433, 449linear programming, 439–446, 451

definition of, 440with more than two linear inequalities,

431–438, 450graph of, 450steps for solving, 437

solution of, 421, 448, 449writing, 448

Systems of quadratic equations, 821–828

with linear equation and quadratic equation, 822–824, 832–833

two quadratic equations, 824–828, 833

TTables

analyzing, 74–77for compound inequalities, 112–113independent and dependent

quantities identified in, 149inverse of function using, 924for linear equations, 185–186for linear functions, 74–77, 95–96of non-linear inverse functions,

911–913sequences as, 252–255for solving linear situations,

comparing equations, and graphs to, 153

technology to complete, 95–96two-way frequency, 581, 609

Termsof polynomial, 704, 705in sequences, 214, 287

formulas to determine, 235–250index of, 238initial, 238nth, 238term number, 238, 239

Significant digits, 167, 203Simple inequalities, 113Simple interest, 296–299, 355–356

formula for, 302rate of change for, 356

Simplifying expressionswith imaginary numbers, 855, 872with negative exponents, 363with negative roots, 854with properties of real numbers,

848–849, 871with square roots, 785

Skewed data distributions, 458, 459, 476, 513, 516

Slope-intercept formof linear equation, 183, 184, 191, 207of literal equation, 191–192

Slopes, common difference as, 291Solutions

of absolute value equations, 125, 127–129, 157

of compound inequalities, 117–120of linear equation, 82, 150, 151of linear functions, 78–80, 82–84of linear inequalities, 106–107of quadratic inequalities, 816, 832of systems of linear equations, 403of systems of linear inequalities, 421,

448, 449Special products, 751–759

of degree 2, 752–755, 783of degree 3, 755–758, 784

Square roots, 763approximate values of, 765–766, 785extracting, 763, 769, 785negative, 763, 785of negative number, 852–854principal (positive), 763, 785simplifying, 785

Squares, difference of two, 752, 783Stacked bar graph, 612Standard deviation, 489–504, 510, 536

calculating and interpreting, 518definition of, 492formula for, 492–493, 518graphing calculator to determine, 495graph of, 497–498, 501, 519

Standard formof linear equations, 183–186, 207of literal equation, 191–192polynomial in, 709quadratic function in, 619, 687

axis of symmetry and vertex of, 777–778

characteristics of parabola given, 696

of systems of linear equations, 377Statistics, 470Stem-and-leaf plot, 508, 520

side-by-side, 508, 520Step functions, 885–894, 923–924

definition of, 891graphing, 891–894, 923greatest integer function (floor

function), 892, 894, 924least integer function (ceiling

function), 893, 894, 924

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imaginary, 863of quadratic function, 640, 691,

772–773completing the square for, 774complex, 863, 872–873connecting x-intercepts of graph

to, 783number of, 797–801, 830quadratic formula to determine,

791, 829See also Roots

Yy-intercept, of equation with two

variables, 205

ZZero, division by, 840Zero Product Property (Converse of

Multiplication Property of Zero), 744, 782

Zeros

WWhole numbers, 838, 869

Xx-intercept

determining, from functions in factored form, 693

of equation with two variables, 205of quadratic functions, 639–640, 691,

693, 783

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