Vocabulary Builder - Salamanca High School€¦ · 4-1 Quadratic Functions and Transformations...

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Vocabulary

Chapter 4 82

4-1 Quadratic Functions and Transformations

Review

1. Circle the vertex of each absolute value graph.

Vocabulary Builder

parabola (noun) puh RAB uh luh

Related Words: vertex, axis of symmetry, quadratic function

Definition: A parabola is the graph of a quadratic function, a function of the form y 5 ax2 1 bx 1 c.

Main Idea: A parabola is symmetrical around its axis of symmetry, a line passing through the vertex. A parabola can open upward or downward.

Use Your Vocabulary

2. Circle each function whose graph is a parabola.

y 5 26x 1 9 y 5 22x

2 2 15x 2 18 y 5 x

2 1 4x 1 4 y 5 16x 2 22

3. Cross out the function(s) whose graph is NOT a parabola.

y 5 5x2 2 3x 1 6 y 5 x 2 3 y 5 2x2 1 6x 2 7 y 5 0.2x 1 7

y

x

y

x

y

x

vertex

parabola

axis ofsymmetry

y y

x

x

82_HSM11A2MC_0401 8282_HSM11A2MC_0401 82 5/18/11 1:50:11 PM5/18/11 1:50:11 PM

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83 Lesson 4-1

Graphing Translations of f(x) 5 x2

Got It? Graph g(x) 5 x2 1 3. How is it a translation of f(x) 5 x2?

Use the graphs of f(x) 5 x2 and g(x) 5 x2 1 3 at the right for Exercises 4 and 5.

4. Circle the ordered pairs that are solutions of g(x) 5 x2 1 3. Underline the ordered pairs that are solutions of f(x) 5 x2.

(23, 0) (23, 9) (0, 23)

(0, 0) (0, 3) (0, 9)

(3, 0) (3, 9) (3, 12)

5. Underline the correct word to complete each sentence.

For each value of x, the value of g(x) 5 x2 1 3 is 3 more / less than

the value of f(x) 5 x2.

The graph of g(x) 5 x2 1 3 is a translation 3 units up / down of the

graph of f(x) 5 x2.

The graph shows f(x) 5 x2 in red / blue and g(x) 5 x2 1 3 in red / blue .

Interpreting Vertex Form

Got It? What are the vertex, axis of symmetry, minimum or maximum, and domain and range of the function y 5 22(x 1 1)2 1 4?

6. Compare y 5 2(x 2 1)2 1 4 with the vertex form y 5 a(x 2 h)2 1 k. Identify a, h, and k.

a 5 h 5 k 5

7. The vertex of the parabola is (h, k) 5 Q , R .

8. The axis of symmetry is the line x 5 .

9. Underline the correct word or symbol to complete each sentence.

Since a is , / . 0, the parabola opens upward / downward .

The parabola has a maximum / minimum value of when x 5 .

10. Circle the domain.

all real numbers x # 21 x # 4 x $ 4

11. Circle the range.

all real numbers x # 21 x # 4 x $ 4

Problem 2

Problem 3

y

xO4 2 2 4

2

2

4

6


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Translation of the Parabola

Chapter 4 84

Writing a Quadratic Function in Vertex Form

Got It? The graph shows the jump of a dolphin. The axis of symmetry is x 5 2, and the height is 7. If the path of the jump passes through the point (5, 5), what quadratic function models the path of the jump?

15. The vertex is Q , R .

16. Substitute h and k in the vertex form f (x) 5 a(x 2 h)2 1 k.

y 5 aQx 2 R21

Problem 5

Using Vertex Form

Got It? What is the graph of f (x) 5 2(x 1 2)2 2 5?

13. Multiple Choice What steps transform the graph of y 5 x2 to y 5 2(x 1 2)2 2 5?

14. Circle the graph of f(x) 5 2(x 1 2)2 2 5.

Problem 4

12. Use one of the functions below to label each graph.

y 5 (x 1 3)2 y 5 x2 2 1 y 5 (x 2 2)2 1 3 y 5 (x 1 1)2 2 2

y

x

yx

y

x

yx

y

x

y

x

y

x

yx

0 2 4 6 8

8

6

4

2

0

y

x

Refl ect across the x-axis, stretch by the factor 2, and translate 2 units to the left and 5 units up.

Stretch by the factor 2 and translate 2 units to the right and 5 units up.

Stretch by the factor 2 and translate 2 units to the left and 5 units down.

Refl ect across the x-axis, stretch by the factor 2, and translate 2 units to the left and 5 units down.


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Lesson Check • Do you UNDERSTAND?

Vocabulary When does the graph of a quadratic function have a minimum value?

19. Circle the parabola that has a minimum value.

20. The graph of y 5 x2 is a parabola that opens upward / downward .

21. The graph of y 5 2x2 is a parabola that opens upward / downward .

22. When does the graph of a quadratic function have a minimum value?

_______________________________________________________________________

_______________________________________________________________________

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

Check off the vocabulary words that you understand.

parabola vertex form quadratic function axis of symmetry

Rate how well you can graph a quadratic function in vertex form.

17. Substitute (5, 5) for (x, y) and solve for a.

18. Write the quadratic function that models the path of the water.

85 Lesson 4-1

y

x

y

x


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Vocabulary

Chapter 4 86

4-2 Standard Form of a Quadratic Function

Review

1. Circle the functions in standard form.

y 5 2x2 2 4x 1 2 y 5 13 (x 2 4) y 5 24x 1 1 y 5 2

23 x 2 5

3 x 1 4

Write each equation in standard form.

2. x 1 2y 5 17 3. 2x 5 5 4. 5 2 x 5 y 1 2

Vocabulary Builder

quadratic (adjective) kwah DRAT ik

Related Words: parabola, vertex, axis of symmetry

Definition: A quadratic function is a function that can be

written in the form y 5 ax2 1 bx 1 c where a 2 0. The graph of a quadratic function is a parabola.

Examples: quadratic functions, y 5 x2, y 5 23x2 1 7, f (x) 5 2x2 1 5x 2 4, g(x) 5 1

2 (x 2 4)2 1 5

Nonexamples: not quadratic functions, y 5 12x2 1 4x 1 5

, x2 1 5x 1 10

3x

Use Your Vocabulary

5. Circle the graphs of quadratic functions.

y

x

y

x

y

x

yx

y ax2 bx c,

a 0

Standard Form of aQuadratic Function


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y

x24 2 4

4

8

4

8

O

87 Lesson 4-2

Finding the Features of a Quadratic Function

Got It? What are the vertex, axis of symmetry, maximum and minimum values, and range of y 5 23x2 2 4x 1 6?

6. Circle the graph of y 5 23x2 2 4x 1 6.

Graphing a Function of the Form y 5 ax2 1 bx 1 c

Got It? What is the graph of y 5 22x 2 1 2x 2 5?

11. The axis of symmetry is x 5 2 b

2a 5 22 ?

5 .

12. Substitute to find the y-coordinate of the vertex.

13. The vertex is Q , R 14. The y-intercept is . The reflection of the y-intercept across

the axis of symmetry is Q , R .

15. Plot the points from Exercises 13 and 14. Draw a smooth curve.

Problem 1

Problem 2

7. Draw and label the axis of symmetry on the graph you circled in Exercise 6.

8. Circle and label the maximum or minimum value on the graph.

9. Circle the range of the function.

y $ 5.0 y # 6.0 all real numbers # 7.3 all real numbers # 9.2

• Th e graph of f (x) 5 ax

2 1 bx 1 c, a 2 0, is a parabola.

• If a . 0, the parabola opens upward. If a , 0, the parabola opens downward.

• Th e axis of symmetry is the line x 5 2

b2a.

• Th e x-coordinate of the vertex is 2

b2a. Th e y-coordinate of the vertex is f Q2

b2aR .

• Th e y-intercept is (0, c).

10. The y-intercept of the graph of f (x) 5 5x

2 2 3x 2 4 is Q , R.

Properties Quadratic Function in Standard Form

HSM11A2MC_0402.indd 87HSM11A2MC_0402.indd 87 6/17/11 1:31:45 PM6/17/11 1:31:45 PM

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Chapter 4 88

Interpreting a Quadratic Graph

Got It? The Zhaozhou Bridge in China is the oldest known arch bridge, dating to 605 a.d. You can model the support arch with the function f (x) 5 20.001075x2 1 0.131148x, where x and y are measured in feet. How high is the arch above its supports?

20. What point on the parabola gives the height of the arch above its supports?

_______________________________________________________________________

21. Find the x-coordinate of the vertex.

x 5 2 b

2a 5 22 ?

5

22. The axis of symmetry of the parabola is x 5 .

23. The length of the bridge is ft.

24. Use the x-coordinate of the vertex to find the y-coordinate.

25. The vertex is about Q , R , so the arch is feet above its support.

Problem 4

Converting Standard Form to Vertex Form

Got It? What is the vertex form of y 5 2x2 1 4x 2 5?

16. Use the justifications at the right to find the vertex.

y 5 Q Rx2 1 Q Rx 1 Q R Write the function in the form y 5 ax2 1 bx 1 c.

x 5 2 b

2a 5 22 ?

5 Find the x-coordinate of the vertex.

Substitute the x-coordinate value into the equation and simplify.

y 5


18. Use the general form of the equation, y 5 a(x 2 h)2 1 k. Substitute for a, h, and k.

y 5 B x 2 Q R R 21 Q R

19. The vertex form of the function is .

Problem 3


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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

024

2

2

6

( 1, 3)

y

x2 2 4

2

4

2

4

O

x =

y = 2(-1)2 - 4(-1) - 3 = 2 + 4 - 3 = 3vertex (-1, 3)

= -1-42(2)

89 Lesson 4-2

• Do you UNDERSTAND?

Error Analysis A student graphed the function y 5 2x2 2 4x 2 3. Find and correct the error.

26. The vertex of y 5 ax2 1 bx 1 c is Q2 b

2a , f Q2 b

2a RR. Find the x- and y-coordinates of the vertex of

y 5 2x2 2 4x 2 3.

27. Find the y-intercept of y 5 2x2 2 4x 2 3.

28. Describe the student’s error and graph the function correctly.

________________________________________________________

________________________________________________________


quadratic standard form vertex axis of symmetry y-intercept

Rate how well you can graph quadratic functions written in standard form.


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Vocabulary

Chapter 4 90

4-3 Modeling With Quadratic Functions

Review

1. Cross out the graphs that are NOT parabolas.

y

x

y

x

y

x

y

x

Vocabulary Builder

model (verb) MAH dul

Main Idea: Modeling is a way of using math to describe a real-world situation.

Definition: A function or equation models an action or relationship by describing its behavior or the data associated with that relationship.

Example: The equation a 5 3g models the relationship between the number of apples, a, and the number of oranges, g, when the number of apples is triple the number of oranges.

Use Your Vocabulary

Draw a line from each description in Column A to the equation that models it in Column B.

Column A Column B

2. The string section of the orchestra has twice y 5 2x 1 1as many violins as cellos.

3. There are two eggs per person with one y 5 100 2 2xextra for good measure.

4. There were 100 shin guards in the closet, and y 5 2xeach player took two.


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91 Lesson 4-3

Writing an Equation of a Parabola

Got It? What is the equation of a parabola containing the points (0, 0), (1, 22), and (21, 24)?

5. Substitute the three points one at a time into y 5 ax2 1 bx 1 c to write a system of equations.

Use (0, 0). 5 aQ R21 bQ R 1 c

Use (1, 22). 5 aQ R21 bQ R 1 c

Use (21, 24). 5 aQ R21 bQ R 1 c

6. Solve the system of equations.

7. The equation of the parabola is y 5 x2 1 x 1 .

Problem 1

Using a Quadratic Model

Got It? The parabolic path of a thrown ball can be modeled by the table. The top of a wall is at (5, 6). Will the ball go over the wall? If not, will it hit the wall on the way up, or the way down?

8. Circle the system of equations you find by substituting the three given points that are on the parabola.

1 5 9a 1 3b 1 c 3 5 a 1 b 1 c 3 5 a 1 b 1 c 2 5 25a 1 5b 1 c 5 5 2a 1 2b 1 c 5 5 4a 1 2b 1 c 3 5 36a 1 6b 1 c 6 5 9a 2 3b 1 c 6 5 9a 1 3b 1 c

9. Now, solve the system of equations.

10. The solution of the system is a 5 , b 5 , c 5 .

11. The quadratic model for the ball’s path is .

12. How can you determine whether the ball goes over the wall? Place a ✓ if the statement is correct. Place an ✗ if it is not.

The value of the model at x 5 5 is at least 6.

The value of the model at x 5 6 is at least 5.

Problem 2

3

5

6

x y

1

2

3


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Chapter 4 92

Using Quadratic Regression

Got It? The table shows a meteorologist’s predicted temperatures for a summer day in Denver, Colorado. What is a quadratic model for the data? Predict the high temperature for the day. At what time does the high temperature occur?

15. Using the LIST feature on a graphing calculator, identify the data that you will enter.

L1 5

L2 5

16. Using a 24-hour clock, write the values for the L1 column.

6 a.m.: 3 a.m.:

9 a.m.: 6 p.m.:

12 p.m.: 9 p.m.:

17. Circle the calculator screen that shows the correct data entry.

6912369

L1 L2 L3637686898576

L2(6) 76

6912151821

L1 L2 L3637686898576

L2(6) 76

6912151821

L1 L2 L3768689857663

L2(6) 63

18. Enter the data from the table into your calculator. Use the QuadReg function. Your screen should look like the one at the right.

Write the quadratic model for temperature.

Problem 3

13. Will the ball go over the wall? Explain.

_______________________________________________________________________

_______________________________________________________________________

14. The value of the model at x 5 6 is less than / greater than value of the model at

x 5 5, therefore the ball was on its way down / up as it approached the wall.

TimePredicted

Temperature (°F)

6 A.M.

9 A.M.

12 P.M.

3 P.M.

6 P.M.

9 P.M.

63

76

86

89

85

76

QuadRegyabc

= ax2 bx c= –.329365= 9.797619= 15.571429


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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

93 Lesson 4-3


Error Analysis Your classmate says he can write the equation of a quadratic function that passes through the points (3, 4), (5, 22), and (3, 0). Explain his error.

22. Graph the points (3, 4), (5, 22), and (3, 0).

23. Underline the correct words to complete the rule for finding a quadratic model.

Two / Three noncollinear points, no two / three of which

are in line horizontally / vertically , are on the graph of exactly

one quadratic function.

24. What is your classmate’s error?

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________


model quadratic model

Rate how well you can use a quadratic model.

19. Use your calculator to find the maximum value of the model. The vertex of the

parabola is Q , R .

20. The high temperature will be °F.

21. At what time will the high temperature occur?

_______________________________________________________________________

y

xO2 2 4 6

2

2

4


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Vocabulary

Chapter 4 94

4-4 Factoring Quadratic Expressions

Review

1. Complete each factor tree.

24

2

3

54

9

Vocabulary Builder

factor (noun) FAK tur

Other Word Forms: factor (verb)

Main Idea: The factors of an expression are similar to the factors of a number.

Definition: The factors of a given expression are expressions whose product equals the given expression. When you factor an expression, you break it into smaller expressions whose product equals the given expression.

Example: The factors of the expression 2x2 2 x 2 10 are 2x 2 5 and x 1 2.

Use Your Vocabulary

2. Circle the prime factors of 24xy.

24 ? x ? y 2 ? 4 ? x ? y 23 ? 3 ? x ? y

3. Circle the prime factors of 54a2b.

54 ? a2 ? b 5 ? 4 ? a2 ? b 2 ? 33 ? a2 ? b

2x2 x 10 (2x 5)(x 2)

expression factors


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95 Lesson 4-4

Factoring ax2 1 bx 1 c when a 5 t1

Got It? What is the expression x2 1 14x 1 40 in factored form?

4. Complete the factor table. Then circle the pair of factors whose sum is 14.

Sum of Factors

Factors of 40 1, 40 2,

5. Circle the expression written as the product of two binomials.

(x 1 1)(x 1 40) (x 1 2)(x 1 20) (x 1 4)(x 1 10) (x 1 5)(x 1 8)

Got It? What is the expression x2 2 11x 1 30 in factored form?

6. Underline the correct word(s) to complete each sentence.

I need to find factors that multiply / sum to 30 and multiply / sum to 211.

At least one of the factors that sum to 211 must be positive / negative .

The two factors that multiply to 30 must both be positive / negative .

7. Circle the factors of 30 that sum to 211.

1 and 30 2 and 15 3 and 10 5 and 6

21 and 230 22 and 215 23 and 210 25 and 26

8. Factor the expression.

x2 2 11x 1 30 5 Qx RQx R

Got It? What is the expression 2x2 1 14x 1 32 in factored form?

9. Rewrite the expression to show a trinomial with a leading coefficient 1.

2x2 1 14x 1 32 5

10. Reasoning You are looking for factors of 232 that sum to 214. Which of the factors has the greater absolute value, the negative factor or the positive factor? How do you know?

_______________________________________________________________________

_______________________________________________________________________

11. Circle the factors of 232 that sum to 214.

21 and 32 22 and 16 24 and 8

1 and 232 2 and 216 4 and 28

12. Write the factored form of the expression.

Problem 1


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Chapter 4 96

Factoring a Perfect Square Trinomial

Got It? What is 64x2 2 16x 1 1 in factored form?

18. Circle the form your answer will have.

(a 1 b)2 (a 2 b)2

Problem 4

Factoring ax2 1 bx 1 c when »a…u 1

Got It? What is the expression 4x2 1 7x 1 3 in factored form? Check your answers.

15. Complete the diagram below.

4x2 7x 3

12

16. Complete the factor pairs of ac. Then circle the pair that sums to 7.

Q1, R Q2, R Q3, R 17. Use your answer to Exercise 16 to complete the diagram below.

Problem 3

Finding Common Factors

Got It? What is the expression 7n2 2 21 in factored form?

13. The GCF of 7n2 and 21 is .

14. Use the Distributive Property to factor the expression.

7n2 1 21 5 Q 1 R

Problem 2

4x2

4x2 (4x)

4x 3

The expressions inside the parentheses must be equal.

7x

4x 3

3

Use the Distributive Property to factor out the GCF, the part inside the parentheses.

33


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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

97 Lesson 4-4


Reasoning Explain how to rewrite the expression a2 2 2ab 1 b2 2 25 as the product of two trinomial factors. (Hint: Group the first three terms. What type of expression are they?)

21. Complete: The first three terms of the expression are a 9.

perfect square trinomial difference of two squares

22. Factor the first three terms of the expression.

23. Rewrite the original expression using the factored form of the first three terms.

24. Complete: The expression you wrote in Exercise 23 is a 9.

perfect square trinomial difference of two squares

25. Circle the expression written as the product of two trinomial factors.

a2 2 2ab 1 b2 (a 2 b)2 2 25 (a 2 b)(225) (a 2 b 2 5)(a 2 b 1 5)


factor of an expression perfect square trinomial difference of two squares

Rate how well you can factor quadratic expressions.

19. Use the justifications to complete each step.

64x2 2 16x 1 1 Write the original expression.

Q xR2 2 16x 1 Q R2 Write the first and third terms as squares.

Q xR22 2Q RQ R x 1 Q R2

Write the middle term as (2ac)x.

20. Write the expression as the square of a binomial.


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Vocabulary

4-5 Quadratic Equations

Review

1. Cross out the equation below that is not a function.

f (x) 5 2x 2 7 y2 5 3x2 2 4x y 5 2x2 1 14x 2 7 g(x) 5 u x3 u

Vocabulary Builder

zero of a function (noun) ZEER oh

Main Idea: Wherever the graph of a function y 5 f (x) intersects the x axis, f (x) 5 0. The value of x at any of these intersection points is called a zero of the function.

Definition: A value of x for which f (x) 5 0 is a zero of the function f (x) .

Example: x 5 2 is a zero of f (x) 5 3x 2 6, because f (2) 5 3(2) 2 6 5 0.

Use Your Vocabulary

Write the zero(s) of each function.

2. 3. 4.

y

xO2 2

2

y

xO 2

2

y

xO2 2

2

Zero(s): Zero(s): Zero(s):

If ab 5 0, then a 5 0 or b 5 0.

Example: If (x 1 7)(x 2 2) 5 0, then (x 1 7) 5 0 or (x 2 2) 5 0.

5. If either x 1 7 5 0 or x 2 2 5 0, circle all of the possible values of x.

27 22 2 27

Key Concept Zero-Product Property

Chapter 4 98


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99 Lesson 4-5

Solving Quadratic Equations by Factoring

Got It? What are the solutions of the quadratic equation x2 2 7x 5 212?

6. The equation is solved below. Write a justification for each step.

x 2 2 7x 5 212 Write the original equation.

x 2 2 7x 1 12 5 0

(x 2 3)(x 2 4) 5 0

x 2 3 5 0 or x 2 4 5 0

x 5 3 or x 5 4

Solving Quadratic Equations With Tables

Got It? What are the solutions of the quadratic equation 4x2 2 14x 1 7 5 4 2 x?

7. Write the equation in standard form.

x2 1 x 1 5 0

8. Enter the equation into your calculator. Use the results to complete the table below.

9. Based on the table, one solution of the equation is x 5 .

10. Another solution occurs between and . Change the x-interval to 0.05. Complete the table.

11. Based on the table, the other solution to the equation is x 5 .

Problem 1

Problem 2

TABLE SETUPTblStart = 0

Tbl = 1IndPnt: Auto AskDePend: Auto Ask

TABLE SETUPTblStart = 0

Tbl = .05IndPnt: Auto AskDePend: Auto Ask

y1

x 0 1 2 3 4

y1

x 0.1 0.15 0.2 0.25 0.3


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Chapter 4 100

Solving a Quadratic Equation by Graphing

Got It? What are the solutions of the quadratic equation x2 1 2x 2 24 5 0?

12. The graph at the right shows the equation. Circle the zeros of the function.

13. The solutions of the quadratic equation

are and .

Using a Quadratic Equation

Got It? The function y 5 20.03x2 1 1.60x models the path of a kicked soccer ball. The height is y, the distance is x, and the units are meters. How far does the soccer ball travel? How high does the soccer ball go? Describe a reasonable domain and range for the function.

14. The graph below shows the function. Circle the point on the graph where the soccer ball is at its highest point and the point where the soccer ball lands. Label each point with its coordinates.

Reasoning Circle the phrase that completes each sentence.

15. The distance the soccer ball travels is the 9.

x-coordinate ofthe vertex

y-intercept

x-coordinate of thepositive zero

y-coordinate of the vertex

16. The maximum height of the soccer ball is the 9.

x-coordinate ofthe vertex

y-intercept

x-coordinate of thepositive zero

y-coordinate of the vertex

17. Underline the correct word to complete each sentence.

The domain should include positive / negative numbers only.

The range should include positive / negative numbers only.

18. Complete.

Domain: # x # Range: # y #

Problem 3

Problem 4

y

x4 8

,

12 16 20 24 28 32 36 40 44 48 52

10

20

,


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Lesson Check

Lesson Check

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101 Lesson 4-5

• Do you know HOW?


Solve the equation x 2 2 9 5 0 by factoring.

19. Circle the phrase that best describes the expression on the left side of the equals sign.

binomial

diff erence of two squares

parabola

quadratic expression

20. Factor the expression on the left side of the equal sign.

21. The solutions of the equation are and .

Vocabulary If 5 is a zero of the function y 5 x2 1 bx 2 20, what is the value of b? Explain.

22. If 5 is a zero of the function then whenever 5 5,

5 0.

23. Substitute for x and solve for .

24. The coefficient b 5 .


zero of a function Zero-Product property

Rate how well you can fi nd the zeros of quadratic equations.


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Vocabulary

Chapter 4 102

4-6 Completing the Square

Review

Draw a line from each expression to its square root.

1. 25x2 x 1 2

2. x2 1 4x 1 4 6x 2 3

3. 36x2 2 36x 1 9 2x 2 5y

4. 4x2 2 20xy 1 25y2 45x

Vocabulary Builder

trinomial (noun) try NOH mee ul

Related Words: perfect square

Definition: A trinomial is an expression consisting of three terms.

Main Idea: You can use perfect square trinomials to solve quadratic equations.

Examples: 4x2 2 7x 1 5, ax2 1 bx 1 c, and 2x 2 5y 1 4z are all trinomials. x2 1 4x 1 4 is a perfect square trinomial because it is the square of the binomial x 1 2.

Use Your Vocabulary

5. Write the number of terms in each expression.

x 1 1 t 2 2 2t 2 6 y 3 p2 2 6p 1 9

6. Put a T next to each expression that is a trinomial. Put an N next to each expression that is not a trinomial.

x2 g 3 1 g 2 4 x2 2 2x 1 5 x2 2 4x

7. Cross out the expression that is NOT a perfect square trinomial.

x2 1 2x 1 1 9x2 2 6x 1 1 4x2 2 4x 2 4 25x2 2 30x 1 9


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103 Lesson 4-6

Determining Dimensions

Got It? The lengths of the sides of a rectangular window have the ratio 1.6 to 1. The area of the window is 2822.4 in2. What are the window dimensions?

8. Circle the equation that represents this situation.

x(1.6x) 5 2822.4 1.6x2 1 x 5 2822.42 (1 1 1.6)x2 5 2822.4

9. The equation is solved below. Write a justification from the box for each step.

Divide each side by 1.6. Simplify. Simplify the left side. Take the square root of each side.

x(1.6x) 5 2822.4 Write the original equation.

1.6x2 5 2822.4

1.6x2

1.6 52822.4

1.6

x2 5 1764

x 5 442

10. One side of the window measures in. The other side measures

1.6Q R , or in.

Problem 2

Solving a Perfect Square Trinomial Equation

Got It? What is the solution of x2 2 14x 1 49 5 25?

11. Use the justifications at the right to solve the equation.

x2 2 14x 1 49 5 25 Write the original equation.

Qx 2 R25 25 Factor the perfect square trinomial.

Qx 2 R 5 4 Take the square root of each side.

x 2 5 or x 2 5 Write as two equations.

x 5 or x 5 Solve for x.

Problem 3

You can turn the expression x2 1 bx into a perfect square trinomial by adding Qb2R

2.

x2 1 bx 1 Qb2R25 Qx 1 b

2R2

Key Concept Completing the Square


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Chapter 4 104

Solving by Completing the Square

Got It? What is the solution of 2x2 2 x 1 3 5 x 1 7?

16. Use the justifications at the right to solve the equation.

2x2 2 x 1 3 5 x 1 7 Write the original equation.

x2 1 x 5 Rewrite so that all terms with x onone side of the equation.

x2 1 ° ¢ x 5 Divide each side by a so thatthe coefficient of x2 is 1.

x2 1 x 5 Simplify.

Qb2R25 Q

2R

2

5 Find Qb2R

2.

x2 1 x 1 5 Add Qb2R

2 to each side.

Qx 1 R25 Factor the trinomial.

x 1 5 Take the square root of each side.

x 5 Solve for x.

Problem 5

Completing the Square

Got It? What value completes the square for x2 1 6x?

13. In the expression, the value b 5 .

14. Circle the expression for the value that completes the square.

62 262 62

2 Q62R

2

15. Complete the square and write the expression as a perfect square.

Problem 4

12. Circle the value that completes the square for x2 1 16x.

4 24 216 64


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Lesson Check

105 Lesson 4-6


How can you rewrite the equation x2 1 12x 1 5 5 3 so that the left side of the equation is in the form (x 1 a)2?

21. Use the justifications at the right to rewrite the equation.


x2 1 12x 5 Rewrite the equation as x2 1 bx 5 c.

x2 1 12x 1 Q2R

2

5 1 Q2R

2

Complete the square.

x2 1 12x 1 5 1 Simplify powers.

x2 1 12x 1 5 Add.

Qx 1 R25 Write as (x 1 a)2 5 c.

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trinomial perfect square trinomial completing the square

Rate how well you can simplify quadratic expressions by completing the square.

Writing in Vertex Form

Got It? What is y 5 x2 1 3x 2 6 in vertex form? Name the vertex and y-intercept.

17. Circle the equation that you can use to complete the square.

y 5 x2 1 3x 1 Q32R

22 6 2 Q3

2R2

y 5 x2 1 3x 2 32 2 6 1 3

2

y 5 x2 1 3x 2 32 2 6 y 5 x2 1 3x 2 Q3

2R2

18. Simplify the equation.

y 5 Qx 1 R22 6 1 5 Qx 1 R2

1


20. The y-intercept is .

Problem 6


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Vocabulary

Chapter 4 106

4-7 The Quadratic Formula

Review

Draw a line from each formula to its description.

1. a 5 s2 area of a circle

2. c 5 2pr circumference of a circle

3. p 5 2(l 1 w) area of a square

4. a 5 pr2 perimeter of a rectangle

Vocabulary Builder

discriminant (noun) dih SKRIM uh nunt

Definition: The discriminant of a quadratic equation in the form ax2 1 bx 1 c 5 0 is the value of the expression b2 2 4ac .

Main Idea: The discriminant helps you determine how many real solutions a quadratic function has.

Use Your Vocabulary

Circle the discriminant of each equation.

5. 2x2 1 (27x) 2 4 5 0

72 2 4(2)24 7 2 4(24) (27)2 2 4(2)(24)

6. 3x2 1 4x 1 2 5 0

4 2 4(3)(2) 12 1 4(3)(2) 42 2 4(3)(2)

7. x2 1 x 2 1 5 0

12 2 4(1)(21) 2 2 4(1)(1) 1(2) 2 (1)(1)

8. 4x2 1 (212x) 1 9 5 0

12 2 4(4)(9) (212)2 2 4(4)(9) (212)2 1 4(4)(9)

discriminant

b2 4ac 0 means 2 real solutions.b2 4ac 0 means 1 real solution.b2 4ac 0 means 0 real solutions.


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107 Lesson 4-7

Using the Quadratic Formula

Got It? What are the solutions to x2 1 4x 5 24? Use the Quadratic Formula.

10. Circle the standard form of the equation.

x2 1 4x 5 24 x2 1 4x 2 4 5 0 x2 1 4x 1 4 5 0

11. Identify the values of a, b, and c.

a 5 b 5 c 5

12. Substitute the values of a, b, and c into the Quadratic Formula. Use the justifications to solve the equation.

x 52b 4" Q R2

2 4Q Rc

2Q R

5

4" Q R22 4Q RQ R

2Q R

54"

5

13. Substitute the value you found in Exercise 12 into the original equation to check your solution.


Q R 2 1 4 Q R 1 4 0 0 Substitute for x.

5 0 ✓ Check for equality.

Problem 1

Write the Quadratic Formula.

Substitute for a, b, and c.

Simplify under the radical.

Simplify.

Key Concept The Quadratic Formula

To solve the quadratic equation ax2 1 bx 1 c 5 0, use the Quadratic Formula.

x 5 2b 4 "b2 2 4ac2a

9. Cross out the value of a that does NOT give a solution to the quadratic formula.

a 5 4 a 5 21 a 5 1 a 5 0


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Using the Discriminant

Got It? What is the number of real solutions of 2x2 2 3x 1 7 5 0?

18. Complete the reasoning model below.

Problem 3

15. Cross out the value that will NOT be substituted into the Quadratic Formula to solve the problem.

21 1 48 2400

16. Substitute values for a, b, and c into the Quadratic Formula and simplify.

x 54" Q R2

2 4Q R Q R2Q R

x 5

4"

< or x <

17. The smallest amount you can charge is for each CD to make a profit

of $100.

WriteThink

a , b , c

b2 4ac 2 4

Find the values of a, b, and c.

Evaluate b2 4ac.

Interpret the discriminant.The discriminant is positive / negative / zero .

The equation has 2 / 1 / 0 real solution(s).

Chapter 4 108

Applying the Quadratic Formula

Got It? Fundraising Your School’s jazz band is selling CDs as a fundraiser. The total profit p depends on the amount x that your band charges for each CD. The equation p 5 2x2 1 48x 2 300 models the profit of the fundraiser. What’s the least amount, in dollars, you can charge for a CD to make a profit of $100?

14. Circle the equation that represents the situation.

0 5 2x2 1 48x 2 200 0 5 x2 1 48x 1 500 0 5 2x2 1 48x 2 400

Problem 2

Using the Discriminant to Solve a Problem

Got It? Reasoning You hit a golf ball into the air from a height of 1 in. above the ground with an initial vertical velocity of 85 ft/s. The function h 5 216t2 1 85t 1 1

12 models the height, in feet, of the ball at time t in seconds. Will the ball reach a height of 120 ft? Explain.

Problem 4


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Lesson Check • Do you UNDERSTAND?

Reasoning For what values of k does the equation x2 1 kx 1 9 5 0 have one real solution? two real solutions?

24. If 9 completes the square, then Qk2R

25 , so k2 5 and k 5 .

25. Place a ✓ if you can use the equation or inequality to solve this problem. Place an ✗ if you cannot.

k2 2 36 5 0 k2 2 36 , 0 k2 2 36 . 0 k2 5 362

26. Now answer the question.

_______________________________________________________________________

_______________________________________________________________________

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quadratic formula discriminant real solutions

Rate how well you can use the quadratic formula to solve problems.

19. Circle the correct strategy to solve the problem.

20. Write the equation in standard form.

t2 1 t 1 5 0

22. The discriminant is positive / negative / zero , so the equation has 2 / 1 / 0

real solutions.

23. The golf ball will / will not reach a height of 120 feet.

Evaluate the discriminant using the values a 5 216, b 5 85, c 5 1

12.

Substitute 120 for h in the equation and evaluate the discriminant to check for real solutions.

Substitute 120 for t in the equation and solve for h.

109 Lesson 4-7

21. Evaluate the discriminant.


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Vocabulary

Chapter 4 110

4-8 Complex Numbers

Review

1. Circle the square root that is not a real number.

!64 !6 2 (2)(4) !4 2 (2)(26) "(25)2

Vocabulary Builder

conjugate (adjective) KAHN juh gut

Related Words: complex numbers, pairs, roots, imaginary solutions

Math Usage: The conjugate of the complex number a 1 bi is a 2 bi .

Main Idea: Complex solutions occur in conjugate pairs of the form a 1 bi and a 2 bi . The product of complex conjugates is always a real number. You can use complex conjugates to simplify division of complex numbers.

Use Your Vocabulary

Write C if the number pairs are complex conjugate or N if they are not.

2. 4 1 3i, 4 2 3i

3. 5 1 !2, 5 2 !2

4. !5 2 !3i, !5 1 !3i

5. 3 1 !5i, 3 1 !25i

Th e imaginary unit i is the complex number whose square is 21. So, i 2 5 21,

and i 5 !21.

For any positive real number a, !2a 5 !21 ? a 5 !21 ? !a 5 i!a.

Note that A!25 B2 5 Ai!5 B2 5 i 2A!5 B2 5 21 ? 5 5 25 (not 5).

Example: !25 5 i!5

Key Concept Square Root of a Negative Real Number


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111 Lesson 4-8

Simplifying a Number Using i

Got It? How do you write the number !212 using the imaginary unit i ?

7. Circle the expression that is equivalent to !212.

!21 ? 4(23) 4!21 ? !3 !21 ? 4 ? 3 22!3

8. Simplify the expression you circled in Exercise 7.

9. Using the imaginary unit i, !212 5 j.

2!3i 4!i 6i 4!3i

Problem 1

Graphing in the Complex Number Plane

Got It? What are the graph and absolute value of 5 2 i?

10. Underline the correct words to complete the sentence.

The graph of 5 2 i is 5 units

left / right and 1 unit up / down .

12. Find the absolute value.

u 5 2 i u 5 ÄQ R21 Q R2

5 Ä 1

5 Ä

Problem 2

Adding and Subtracting Complex Numbers

Got It? What is the sum (7 2 2i) 1 (23 1 i)?

Problem 3

6. Use !21 5 i to complete each equation.

!22 5 !2 !23 5 iÅ !26 5 Å 5 i!8

11. Graph the point.

imaginary axis

real axis24 2 4

2i

4i

2i

4i

Use the Distance Formula.

Simplify powers.

Add.


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Chapter 4 112

Dividing Complex Numbers

Got It? What is the quotient 5 2 2i3 1 4i?

15. Circle the first step in simplifying the fraction.

Find the complex conjugate of 5 2 2i. Find the complex conjugate of 3 1 4i.

Find the absolute value of 5 2 2i. Find the absolute value of 3 1 4i.

16. Cross out the expression that is NOT equivalent to the quotient.

15 2 20i 2 6i 1 8i 2

9 2 12i 1 12i 2 16i 2 15 2 26i 1 8i

2

25 25 1 10i 2 10i 2 16i

2

9 2 12i 1 12i 2 16i 2

17. Simplify.

18. 5 2 2i3 1 4i 5

Problem 5

Multiplying Complex Numbers

Got It? What is the product (7i)(3i)?

14. Complete the solution. Justifications are given.

(7i)(3i) Write the original expression.

i 2 Multiply.

Q RQ R Substitute 21 for i 2.

Simplify.

Problem 4

13. The sum is found below. Write the justification for each step.

(7 2 2i) 1 (23 1 i) Write the original expression.

7 1 (22i 2 3) 1 i

7 1 (23 2 2i) 1 i

(7 2 3) 1 (22i 1 i)

4 2 i

Property

Property

Property


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Lesson Check

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Now Iget it!

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0 2 4 6 8 10

(4 - 7i) (4 + 7i) = 16 + 28i - 28i + 49i2 = 16 - 49 = -33

113 Lesson 4-8


Error Analysis Describe and correct the error made in simplifying the expression (4 2 7i)(4 1 7i).

21. Simplify the expression.

22. Explain the error shown above.

_______________________________________________________________________

_______________________________________________________________________


imaginary number complex number complex conjugates

Rate how well you can fi nd complex-number solutions to quadratic equations.

Factoring using Complex Conjugates

Got It? What are the factors of each expression?

19. 5x 2 1 20

What is the GCF of 5 and 20?

Write as a product using the GCF.

Rewrite x 2 1 4 as a2 1 b2. a 5

b 5

Use a2 1 b2 5 (a 1 bi)(a 2 bi).

What are the factors of 5x 2 1 20?

20. x 2 1 81 Rewrite x

2 1 81 in terms of a2 1 b2.

Use a2 1 b2 5 (a 1 bi)(a 2 bi).

Problem 6


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Vocabulary

Chapter 4 114

4-9 Quadratic Systems

Review

Write T for true or F for false.

1. The solution of system y 5 3x 1 2 and y 5 5x is the point where the two lines intersect.

2. The solution of a system of 2 linear equations has at most 2 points of intersection.

3. The solution of a system of inequalities is the point where the lines intersect with the y-axis.

4. The solution of a system of inequalities is the region where the graphs of the inequalities overlap.

Vocabulary Builder

Quadratic-Linear System (noun) kwah DRAT ik LIN ee ur SIS tum

Related Words: System of equations, system of inequalities.

Main Idea: A system of equations can include an equation with a graph that is not a line. Such a system can have more than one solution.

Definition: A quadratic-linear system is a system of one quadratic equation and one linear equation. The system can have two, one, or no solutions (points of intersection).

Use Your Vocabulary

5. Cross out the graph that does NOT illustrate a quadratic-linear system.


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115 Lesson 4-9

Solving a Linear-Quadratic System by Graphing

Got It? What is the solution of the system? e y 5 x 2 1 6x 1 9

y 5 x 1 3

6. Complete the table of values for 7. Use the points from the table toboth equations. graph the two equations.

x2 6x 9 x 3x

4

3

2

1

0

1

y

xO24 2 4

2

2

4

6

8. The solutions are Q , R and Q , R .

9. Substitute into both equations to check the solutions.

Problem 1

Solving Using Substitution

Got It? What is the solution of the system? e y 5 2x 2 2 3x 1 10

y 5 x 1 5

10. Use the justifications at the right to solve the system.

x 1 5 x

2 1 x 1 Substitute x 1 5 for y in the

quadratic equation.

x 2 1 x 1 5 0 Write in standard form.

Qx 1 RQx 1 R 5 0 Factor.


x 5 S y 5 1 5 5 Substitute for x in y 5 x 1 5.

x 5 S y 5 1 5 5

Problem 2


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Lesson Check

y

xO24 2

2

6

8

Chapter 4 116

• Do you know HOW?

Solve the system by substitution. e y 5 x 2 2 2x 1 3

y 5 x 1 1

11. Check the solutions.

Solution 1 Q , R Solution 2 Q , R y 5 2x

2 2 3x 1 10 y 5 2x 2 2 3x 1 10

5 2Q R22 3Q R 1 10 5 2Q R2

2 3Q R 1 10

5 5

Solving a Quadratic System of Equations

Got It? What is the solution of the system? e y 5 x 2 2 4x 1 5

y 5 2x 2 1 5

12. Circle the graph of the system. Each graph shows the standard viewing window.

Solving a Quadratic System of Inequalities

Got It? What is the solution of this system of inequalities? e y K 2x 2 2 4x 1 3

y S x 2 1 3

Th e graph at the right shows the boundaries of the inequalities.

14. Shade the region that represents y # 2x 2 2 4x 1 3.

15. Shade the region that represents y . x 2 1 3 in

another color.

16. Outline the region that represents the solution of the system of inequalities.

Problem 3

Problem 4

13. Use the graph you circled. Circle the solution of the system.

(22, 1) (21, 4) (0, 5) (1, 2) (2, 1) (3, 24)


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Lesson Check

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117 Lesson 4-9


Reasoning How many points of intersection can you have between a linear function and a quadratic function? Draw graphs to justify your answers.

18. If possible, draw linear function and a quadratic function with the number of intersections specified.

0 points of intersection 1 point of intersection 2 points of intersection

y

xO2 2

2

2

y

xO2 2

2

2

y

xO2 2

2

2

19. Circle the number(s) of points of intersection you can have between a linear function and a quadratic function.

0 1 2 3 4


quadratic-linear system system of equations system of inequalities

Rate how well you can solve and graph systems of equations and inequalities.

17. Complete the solution. Justifications are given.

x 1 1 5 x 2 2 2x 1 3 Substitute x 1 1 for y.

0 5 x 2 x 1 Addition property of equality.

0 5 Qx 2 RQx R Factor.


y 5 1 1 1 5 or y 5 1 1 5 Substitute for x in y 5 x 1 1 and solve for y.

The solutions are Q1, R and Q , R .


Vocabulary Builder - Salamanca High School€¦ · 4-1 Quadratic Functions and Transformations...

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Transcript of Vocabulary Builder - Salamanca High School€¦ · 4-1 Quadratic Functions and Transformations...