NAME: …pierrekac.weebly.com/uploads/3/0/4/1/30416280/1._notes_3.6.2_answer_key.pdfLesson 3.6.2:...

Name: ___________________________________ # ___________ Honors Coordinate Algebra: Period _____________ Ms. Pierre Date: ______________

3.6.2 Constructing Functions from Graphs & Tables

Warm Up

LINEAR AND EXPONENTIAL FUNCTIONSLesson 6: Building Functions

NAME:

CCGPS Coordinate Algebra Teacher Resource Binder U3-330

© Walch Education

Lesson 3.6.2: Constructing Functions from Graphs and TablesWarm-Up 3.6.2Read the scenario and use the table to complete problem 1. For problems 2 and 3, read each scenario and create a graph of each linear function.

1. The students in Mr. Jacobs’ class record their age in months and their height in centimeters. The data from 10 of the students is in the table. Create a graph of the students’ ages and heights.

Age in months Height in centimeters162 146165 149164 180158 126156 140166 183160 176161 193167 184156 125

2. Elliot uses the equation f (x) = 65x to estimate the number of miles traveled after any number of hours, x, during a recent road trip.

3. A store is having a sale. T-shirts are $5, and the store is offering $3 off any purchase. The total purchase cost of any number of T-shirts can be represented by the equation f (x) = 5x – 3.


NAME:


© Walch Education







NAME:


© Walch Education







NAME:


© Walch Education







NAME:


© Walch Education







Instruction

CCGPS Coordinate Algebra Teacher Resource Binder © Walch EducationU3-335

Prerequisite Skills

This lesson requires the use of the following skills:

• graphing points on the coordinate plane

• understanding independent and dependent quantities

IntroductionTables and graphs can be represented by equations. Data represented in a table can either be analyzed as a pattern, like the data presented in the previous lesson, or it can be graphed first. The shape of a graph helps identify which type of equation can be used to represent the data. Data that is represented using a linear equation has a constant slope and the graph is a straight line. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is the point at which the graph intersects the y-axis. To find the equation of a line, first find the slope, then replace x and y in the general equation with x and y from an ordered pair on the graph. Then, solve the equation for b.

The slope of a line is the change in the independent variable divided by the change in the dependent

variable, and describes the rate of change of the variables. Using the points (x1, y

1) and (x

2, y

2), the slope

can be calculated by finding y y

x x2 1

2 1

( )( )

−−

. The slope is similar to the common difference between terms,

except the slope can be found between points that have x-values that are more than one unit apart. The

slope allows us to compare the change between any two points on a graph or in a table. An exponential

equation has a slope that is constantly changing, either increasing or decreasing. Graphically,

exponential equations are a curve. The general form of an exponential equation is y= ab x, where a and

b are real numbers, and b is the common ratio. The values of a and b change the shape of the graph of

an exponential function. On the following pages, you’ll find four examples of functions with different a

and b values.


Instruction


Prerequisite Skills

This lesson requires the use of the following skills:

• graphing points on the coordinate plane

• understanding independent and dependent quantities

IntroductionTables and graphs can be represented by equations. Data represented in a table can either be analyzed as a pattern, like the data presented in the previous lesson, or it can be graphed first. The shape of a graph helps identify which type of equation can be used to represent the data. Data that is represented using a linear equation has a constant slope and the graph is a straight line. The general form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept. The y-intercept is the point at which the graph intersects the y-axis. To find the equation of a line, first find the slope, then replace x and y in the general equation with x and y from an ordered pair on the graph. Then, solve the equation for b.

The slope of a line is the change in the independent variable divided by the change in the dependent

variable, and describes the rate of change of the variables. Using the points (x1, y

1) and (x

2, y

2), the slope

can be calculated by finding y y

x x2 1

2 1

( )( )

−−

. The slope is similar to the common difference between terms,

except the slope can be found between points that have x-values that are more than one unit apart. The

slope allows us to compare the change between any two points on a graph or in a table. An exponential

equation has a slope that is constantly changing, either increasing or decreasing. Graphically,

exponential equations are a curve. The general form of an exponential equation is y= ab x, where a and

b are real numbers, and b is the common ratio. The values of a and b change the shape of the graph of

an exponential function. On the following pages, you’ll find four examples of functions with different a

and b values.


Instruction


© Walch Education

a > 0 and b > 1

f (x) = 2 x

-10 -8 -6 -4 -2 0 2 4 6 8 10

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a > 0 and 0 < b < 1

f (x) = (0.5) x

-10 -8 -6 -4 -2 0 2 4 6 8 10

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10


Instruction


© Walch Education

a > 0 and b > 1

f (x) = 2 x

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

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a > 0 and 0 < b < 1

f (x) = (0.5) x

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Instruction


a < 0 and b > 1

f (x) = (–1) • 2 x

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a < 0 and 0 < b < 1

f xx

( ) 11

2= −

⎛⎝⎜

⎞⎠⎟

-10 -8 -6 -4 -2 0 2 4 6 8 10

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Instruction


a < 0 and b > 1

f (x) = (–1) • 2 x

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

-8

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-2

2

4

6

8

10

a < 0 and 0 < b < 1

f xx

( ) 11

2= −

⎛⎝⎜

⎞⎠⎟

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

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Instruction


© Walch Education

Data that follows an exponential pattern has a common ratio between the dependent quantities. When looking for the common ratio, be sure to verify that the x-values, or independent quantities, are each one unit apart. The common ratio, b, can be used to write an equation to represent the exponential pattern. Find the value of the equation at x = 0, f (0). The general equation of the function is f (x) = f (0) • bx.

Key Concepts

• The graph of a linear equation is a straight line.

• Linear equations have a constant slope, or rate of change.

• Linear equations can be written as functions.

• The general form of a linear function is f (x) = mx + b, where m is the slope and b is the y-intercept.

• The slope of a linear function can be calculated using any two points, (x1, y

1) and (x

2, y

2): the

formula is y y

x x2 1

2 1

( )( )

−−

.

• The y-intercept is the point at which the graph of the equation crosses the y-axis.

• Exponential equations have a slope that is constantly changing.

• Exponential equations can be written as functions.

• The general form of an exponential function is f(x) = ab x, where a and b are real numbers.

• The graph of an exponential equation is a curve.

• The common ratio, b, between independent quantities in an exponential pattern, and the value of the equation at x = 0, f(0), can be used to write the general equation of the function: f(x) = f (0) • b x.

Common Errors/Misconceptions

• looking for a common difference between dependent quantities when the independent values are not one unit apart

• incorrectly calculating the slope

• trying to match an exponential graph to a linear equation

• trying to match a linear graph to an exponential equation


Instruction


© Walch Education


Key Concepts






1) and (x

2, y

2): the

formula is y y

x x2 1

2 1

( )( )

−−

.













Instruction


© Walch Education


Key Concepts






1) and (x

2, y

2): the

formula is y y

x x2 1

2 1

( )( )

−−

.












Example 1


Instruction


Guided Practice 3.6.2Example 1

Determine the equation that represents the relationship between x and y in the graph below.

-5 -4 -3 -2 -1 0 1 2 3 4 5

-9

-8

-7

-6

-5

-4

-3

-2

-1

1. Determine which type of equation, linear or exponential, will fit the graph.

The graph of a linear equation is a straight line, and a graph of an exponential equation is a curve. An exponential equation can be used to represent the graph.

2. Identify at least three points from the graph, with one of the points at x = 0.

From the graph, three points are: (0, –1), (1, –3), and (2, –9).

Example 2


Instruction


Example 2

Determine the equation that represents the relationship between x and y in the graph below.

-5 -4 -3 -2 -1 0 1 2 3 4 5

-4

-3

-2

-1

1

2

3

4

5

1. Determine which type of equation, linear or exponential, will fit the graph.

The graph of a linear equation is a straight line, and a graph of an exponential equation is a curve. A linear equation can be used to represent the graph.

2. Identify at least three points from the graph.

From the graph, three points are: (0, –4), (1, –1), and (2, 2).

3. Find the slope of the line, using any two of the points.

The slope of the line is y y

x x2 1

2 1

( )( )

−−

for any two points (x1, y

1) and (x

2, y

2).

Using the points (0, –4) and (1, –1), the slope is 1 4

1 03

( )( )( )

− − −−

= .

Example 3


Instruction


Example 3

A clothing store discounts items on a regular schedule. Each week, the price of an item is reduced. The prices for one item are in the table below. Week 0 is the starting price of the item.

Week Price in dollars ($)0 100.00 1 60.00 2 36.00 3 21.60

Determine a linear or exponential equation that represents the relationship between the week and the price of the item.

1. Create a graph of the data.

Let the x-axis represent the week, and the y-axis represent the price in dollars.

0 1 2 3 4 5 6 7 8 9 10

10

20

30

40

50

60

70

80

90

100

Week

Pric

e in

dol

lars

($)

Guided Practice

1. )

2.)

3.)


NAME:


Practice 3.6.2: Constructing Functions from Graphs and TablesDetermine a linear or exponential equation that represents the relationship between x and y in each graph or table that follows.

1.

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-4 -2 0 2 4 6 8 10

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continued


NAME:



1.

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-4 -2 0 2 4 6 8 10

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continued


NAME:



1.

-10 -8 -6 -4 -2 0 2 4 6 8 10

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-4 -2 0 2 4 6 8 10

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continued


NAME:


© Walch Education

5.

-10 -8 -6 -4 -2 0 2 4 6 8 10-2

2

4

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16

18

6. x y0 10 1 40 2 160 3 640 4 2560

7. x y0 162 133 11.55 8.5

8. x y0 –121 –44 205 28

9. x y0 –21 –142 –983 –6864 –4802

10. x y0 –71 –562 –4483 –3584

Independent Practice

a.) What is the slope of the equation that represents the relationship?

b.) Should the relationship be represented using a linear or exponential equation?


NAME:


Problem-Based Task 3.6.2: Interior Angles in PolygonsJulia is studying the sum of interior angles in polygons. She creates polygons with 3, 5, 6, 7, and 9 sides, and records the sum of the interior angles in each polygon in the table below.

Number of sides Sum of interior angles, in degrees3 1805 5406 7207 9009 1260

Is there a relationship between the number of sides of a polygon and the sum of the interior angles? Find an equation to determine the sum of the interior angles of a polygon with any number of sides.

c.) Justify your answer. This graph is a line, with a constant rate of change, and the relationship should be represented using a linear equation.

d.) Write an equation to represent the sum of the interior angles for any polygon.

Homework

1. )

2.)


NAME:



1.

-10 -8 -6 -4 -2 0 2 4 6 8 10

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-4 -2 0 2 4 6 8 10

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continued


NAME:



1.

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-4 -2 0 2 4 6 8 10

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14

continued


NAME:


© Walch Education

5.

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2

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18

6. x y0 10 1 40 2 160 3 640 4 2560

7. x y0 162 133 11.55 8.5

8. x y0 –121 –44 205 28

9. x y0 –21 –142 –983 –6864 –4802

10. x y0 –71 –562 –4483 –3584

3.) 4.)


NAME:


© Walch Education

5.

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2

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6. x y0 10 1 40 2 160 3 640 4 2560

7. x y0 162 133 11.55 8.5

8. x y0 –121 –44 205 28

9. x y0 –21 –142 –983 –6864 –4802

10. x y0 –71 –562 –4483 –3584


NAME:


© Walch Education

5.

-10 -8 -6 -4 -2 0 2 4 6 8 10-2

2

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6. x y0 10 1 40 2 160 3 640 4 2560

7. x y0 162 133 11.55 8.5

8. x y0 –121 –44 205 28

9. x y0 –21 –142 –983 –6864 –4802

10. x y0 –71 –562 –4483 –3584

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