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CP Algebra 2

Unit 1: Algebra Review

Name: _________________________Period_____________

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Learning Targets:

Section A Algebraic

Expressions & Functions

1. I can simplify and evaluate algebraic expressions.

2. I can identify functions and evaluate functions written in function notation.

3. I can simplify and rationalize square roots.

Section B Equations & Inequalities

4. I can solve equations, solve basic inequalities, and graph inequalities on a number line.

5. I can solve compound inequalities and absolute value inequalities.

Section C Graphing

6. I can graph linear functions.

7. I can graph and apply piecewise functions.

Section D Writing

Equations

8. I can write equations of lines and piecewise functions.

9. I can apply equations of lines to real world problems.

10. I can use linear regression to solve real world problems.

Section E Systems 11. I can solve systems of equations by substitution, elimination, and graphing.

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Section A: Algebraic Expressions & Functions After this lesson and practice, I will be able to …

! simplify and evaluate algebraic expressions. (LT 1) ! identify functions and evaluate functions written in function notation. (LT 2) ! simplify and rationalize square roots. (LT 3)

---------------------------------------------------------------------------------------------------------------------------------- One of the most important skills you will need this year is evaluating algebraic expressions. Example 1: Evaluate the expression. a. Evaluate 3𝑏 − 𝑎𝑐 + 4𝑏 for 𝑎 = 2, 𝑏 = −4, and 𝑐 = 3. b. Evaluate −𝑥! + 4(𝑥𝑦 − 5) for 𝑥 = 3 and 𝑦 = −2. c. Evaluate 7𝑟 + 2(𝑠)! for 𝑟 = 3 and 𝑠 = −5. d. Evaluate 2 3𝑦 − 5𝑦 for 𝑦 = −1.

We will also need to simplify expressions. Remember that you can always combine like terms, or terms that have the same _____________ and _________________.

Example 2: Simplify the algebraic expression by using the distributive property and by combining like terms. a. −2 𝑚 − 𝑛 +𝑚! + 𝑛 b. −3 𝑎 + 𝑏 − (2𝑏 − 6𝑎)

Be careful when substituting a negative value when squaring:

-32 = (-3)2 =

Be careful when subtracting a polynomial 4 – (x + 9) =

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LT 2 (Functions) We will spend a lot of time graphing functions this year, so we will need to identify domain and range. ★ Domain: ★ Range: Example 3: Express the relation !{(2,−4),(4,−4),(−3,1),(0,−2),(−1,3),(−3,−1)}as a mapping diagram.

Domain: _______________________ Range: __________________________

A relation simply describes a relationship between x values and y values. A function is a special type of relation, where each input (or x value) has ________________________ output. Example 3 Continued: Is the relation from Example 3 a function? Why or why not?

Example 4: Determine the domain and range of the relation below. Express your answer using set notation. Then determine if the relation is a function.

Domain: _____________________ Domain: _____________________ Domain: _____________________ Range: ______________________ Range: ______________________ Range: ______________________ Function or not: ______________ Function or not: ______________ Function or not: ______________

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Recall function notation:

f(x)

f(5) means

5f(3) means

5 is called a ______________________

Example 5: Find !!f (2), f (−3), and !!f (0.5) for each of the following functions. Show all work. a. !!f (x)=2x +4 b. !!f (x)=2x

2 −3

f (2)=

f (−3)=

f (0.5)=

f (2)=

f (−3)=

f (0.5)=

LT 3 (Square Roots) The last skill we need to refresh in this section has to do with radicals. Let’s start by reviewing the perfect squares … X

0

1

2

3

4

5

6

7

8

9

X2

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48 Rational Number – Irrational Number - Radicand - Denominator – Example 8: Simplify completely. (Remember: To simplify square roots, get all perfect squares out of the radicand.)

a. 75 b. 128 c. 2 28 d. 4 9 To rationalize a denominator, multiply the numerator and denominator by a radical that will simplify so there is not a radical in the denominator.

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Example 9: Simplify completely. Rationalize the denominator if necessary.

a. 25 b.

63 c.

142

Example 10: Simplify completely. Rationalize the denominator if necessary.

a. 62 3

b. 4−2 38 c.

10+15 210

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Topic B: Equations and Inequalities After this lesson and practice, I will be able to …

! solve equations, solve basic inequalities and graph inequalities on a number line. (LT 4) ! solve compound inequalities and absolute value inequalities. (LT 5)

---------------------------------------------------------------------------------------------------------------------------------- Example 1: Solve each equation. Leave answers as reduced fraction is needed (no decimals). a. 1+ 3𝑛 = −9− 2𝑛 b. 17− 8𝑏 = 4− 8𝑏 + 6 c. 4𝑛 + 6 = 4𝑛 + 6 e. − 𝑛 − 3 + 5 = 7𝑛 − 2 3𝑛 + 3 Example 2: Solve each literal equation for the indicated variable.

a. !!!!+ 𝑏 = 𝑎, solve for 𝑥 b. 𝑅 𝑟! + 𝑟! = 𝑟!𝑟!, solve for 𝑟!

Example 3: Solve by writing an equation. Use labels when writing the final answer. a. Find 4 consecutive odd integers with a sum of 184.

When you get something that makes sense … When it doesn’t make sense …

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b. A passenger plane made a trip to Dublin and back. The trip there took seven hours and the trip back took three hours. It averaged 200 km/h faster on the return trip than on the outbound trip. Find the passenger planes average speed on the outbound trip.

c. A passenger plane left Los Angeles and flew east at an average speed of 290mph. A cargo plane left flying in the opposite direction with an average speed of 370mph. When will they be 3300 miles apart?

LT 4 Continued – Solving Inequalities Solving an inequality works the same way as solving an equation, but our answers will be x > something or x < something. Example 4: Solve each inequality and graph the solution. a. 12 ≥ 24𝑥 b. 14− 4𝑦 < 38

c. 2 3−𝑚 − 7 < 9

When graphing on a number line:

When multiplying or dividing by a

negative number:

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Example 5: Solve each problem by writing an inequality. Use labels when writing the final answer. a. The length of a picture frame is 3 in. greater than the width. The perimeter is less than 52 in. Describe the dimensions of the frame.

b. Find the lesser of two consecutive integers with a sum greater than 16.

LT 5 (Compound and Absolute Value Inequalities) Graphing OR versus AND OR AND Example 6: Solve each compound inequality. Graph the solution.

a. 4𝑥 < 16 or 12𝑥 > 72 b. 2𝑥 − 1 > −11 and 9𝑥 + 2 < 20

c. −50 ≤ 7𝑘 + 6 < −8

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Example 7: Solve each inequality and graph the solution. a. |𝑥 + 3| > 9

b. |𝑥 – 5| + 2 ≥ 10

c. 3|2𝑥 - 3| ≤ 15

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Topic C: Graphing After this lesson and practice, I will be able to …

! graph linear functions. (LT 6) graph and apply piecewise functions. (LT 7) ---------------------------------------------------------------------------------------------------------------------------------------------- We will be doing a lot of graphing this year! So let’s start by reviewing how to graph linear functions.

x-intercept y-intercept Example 1: Determine the x- and y-intercepts of a linear equation. Write as ordered pairs.

a. !!y =3x +9 b. c. !!2x −4 y =12

x-int __________ x-int __________ x-int __________

y-int __________ y-int __________ y-int __________ There are several ways you can write the equation of a line: Slope – Intercept Form: Standard Form: Point – Slope Form: Example 2: Graph linear equations written in slope-intercept form, standard form, or point-slope form. ________________________ Form

a. !!y = −3x +4

b. !!x = −6

c. !!y =7

d. !!y = 43 x −6

e. !!y = −27 x −1

!!y −2=3(x −1)

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________________________ Form

a. !!x +5y =10

b. !!3x −4 y =16

c. !!−2x +4 y =12

d. !!𝑥 + 𝑦 = 2

________________________ Form

a. !!y +1=2(x −5)

b. !!y −5= −3(x +2)

c. !!y +3= 34(x +1)

d. !!y −6= −12(x −4)

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LT 7 (Graph Piecewise Functions) Sometimes situations cannot be accurately modelled using a single equation. In some instances, it is helpful to define separate functions based on specified domain values. For such situations, you can utilize ___________________________________. Example 3:

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, if 0( )

2, if 0x x

f xx x

<!= "

+ ≥$

f (x) =

x, if x < −414x − 3, if − 4 ≤ x ≤ 4

x − 6, if x > 4

#

$%%

&%%

( 1) _____

(0) _____

(1) _____

(4) _____

f

f

f

f

− =

=

=

=

f (−8) = ______

f (2) = ______

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Applications of Piecewise Functions Example 4: A local zoo charges admission to groups according to the following policy. Groups of fewer than 50 people are charged a rate of $35 per person, while groups of 50 or more people are charged a reduced rate of $30 per person. a) Write a mathematical model expression the amount a group will be charges for admission as a function of its size.

b) Sketch the function. Label the x and y axis.

What would the price be for a group of: c) 20 people? ________________ d) 49 people? ________________

e) 50 people? ________________ f) 51 people? ________________

f (x) =

!

"

###

$

###

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Topic D: Writing Equations After this lesson and practice, I will be able to …

! write equations of lines and piecewise functions. (LT 8) ! apply equations of lines to real world problems. (LT 9) ! use linear regression to solve real world problems. (LT 10)

---------------------------------------------------------------------------------------------------------------------------------- In this lesson we’ll be doing the reverse of the previous lesson: we’ll write equations based on graphs or based on information like points or intercepts. Example 1: Write the equation of the line in STANDARD FORM with given slope and passing through the given point.

a. Slope = 3; (1, 5) b. Slope = 1/2; (22, -10) Example 2: Write in SLOPE-INTERCEPT FORM the equation of the line passing through the points.

a. (-10, 3) and (-2, -5) b. (1,0) and (5,5) Example 3: Write the equation of the graphed line.

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Example 4: Write a piecewise function for the following graphs. a. b. LT 9 (Applying Equations of Lines) Example 5: Suppose an airplane descends at a rate of 300 ft/min from an elevation of 7000 ft.

a. Write an equation for the plane’s elevation as a function of the time it has been descending.

b. Find and interpret the intercepts in this situation.

Example 6: A candle is 6 in. tall after burning for 1 hr. After 3 hr., it is 5.5 in. tall. a. Write an equation for the height of the candle as a function of the time it’s been burning.

b. If the candle is 3.75 in. tall, how many hours has the candle been burning?

f (x) =

!

"

###

$

###

f (x) =

!

"

###

$

###

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LT 10 (Linear Regression) You might remember learning about the “Line of Best Fit” in Algebra 1. The line of best fit shows us the relationship between two variables, and we can find it using our calculators …

★Notes on using the graphing calculator:

Example 7: The table compares the relative humidity to the apparent temperature in a room that has a room temperature of 72 degrees Fahrenheit.

a. Graph the data in your calculator.

b. Write the equation for the line of best fit. Round to three decimal places.

c. What is the real-world meaning of your slope?

d. Using your equation from part b, what would you expect the apparent temperature to be if the relative humidity were 73%?

Relative Humidity

(%)

Apparent Temperature

(degrees Fahrenheit)

0 64

10 65

20 67

30 68

40 70

50 71

60 72

70 73

80 74

90 75

100 76

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Topic E: Solving Systems After this lesson and practice, I will be able to …

! solve systems of equations by substitution, elimination, and graphing. (LT 11) ---------------------------------------------------------------------------------------------------------------------------------- Graphing When graphing to solve a system of equations, the solution is where the lines INTERSECT. If the lines coincide, state the solution as ALL POINTS ON ….. (state the equation of the line(s)). If the lines do not intersect there is NO SOLUTION.

1. 𝑦 = 𝑥 + 3𝑦 = 2𝑥 + 3 ______________ 2.

𝑦 = 𝑥𝑦 = − !

!𝑥 + 3 ______________

Substitution Substitution is a great method for solving when a variable is already isolated (or close to being isolated).

1. 𝑦 − 𝑥 = −1

−4𝑥 + 3𝑦 = −2 ______________ Check:

2. −6 = 3𝑥 − 6𝑦4𝑥 = 4+ 5𝑦 ______________ Check:

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Elimination Elimination is a popular method to use when variables line up nicely …

1. 3𝑥 + 2𝑦 = 63𝑥 + 3 = 𝑦 ______________ Check:

2. 5𝑥 − 𝑦 = 42𝑥 − 𝑦 = 1 ______________ Check:

Use Any Method!

1. 𝑚 + 2𝑛 = 53𝑚 + 5𝑛 = 11 ______________ 2.

2𝑥 + 3𝑦 = 11 5𝑥 − 2𝑦 = −20 ______________