Algebra I - NJCTLcontent.njctl.org/.../non-linear-functions-2-2015-11-04.pdf2015/11/04 · ax2 + bx + c = 0 , where a is not equal to 0. The form ax2 + bx + c = 0 is called the standard

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Algebra I

Quadratic & Non-Linear Functions

2015-11-04

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3

Table of ContentsClick on the topic to go to that section

• Explain Characteristics of Quadratic Functions

• Graphing Quadratic Functions in Standard Form

• Transforming and Translating Quadratic Functions

• Key Terms

• Graphing Quadratic Functions in Vertex and Intercept Form

• Comparison of Types of Functions

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Key Terms

Return to Tableof Contents

5

Axis of symmetry: The vertical line thatdivides a parabola intotwo symmetrical halves

Axis of Symmetry

6

Maximum: The y-value of thevertex if a < 0 and the parabola opens downward.

Minimum: The y-value of thevertex if a > 0 and theparabola opens upward.

Parabola: The curve resultof graphing aquadratic equation.

(+ a) Max

Min

(- a)

Parabolas

7

Quadratic Equation: An equation that can be written in the standard form ax2 + bx + c = 0. Where a, b and c are real numbers and a does not = 0.

Quadratic Function:Any function that can be written in the form y = ax2 + bx + c. Where a, b and c are real numbers and a does not equal 0.

Vertex: The highest or lowest point ona parabola.

Zero of a Function: An x value that makes the function equal zero.

Quadratics

8

Explain Characteristics of Quadratic Equations


9

A quadratic equation is an equation of the form ax2 + bx + c = 0 , where a is not equal to 0.

The form ax2 + bx + c = 0 is called the standard form of the quadratic equation.

The standard form is not unique. For example, x2 – x + 1 = 0 can be written as the equivalent equation –x2 + x – 1 = 0.

Also, 4x2 – 2x + 2 = 0 can be written as the equivalent equation 2x2 – x + 1 = 0. Why is this equivalent?

Quadratics

10

Practice writing quadratic equations in standard form. (Simplify if possible.)

Write 2x2 = x + 4 in standard form.

Writing Quadratic Equations

Answ

er

11

Write 3x = –x2 + 7 in standard form.


Answ

er

12

Write 6x2 – 6x = 12 in standard form.


Answ

er

13

Write 3x – 2 = 5x in standard form.


Answ

er

14

Characteristics of Quadratic Functions

1. Standard form is y = ax2 + bx + c, where a ≠ 0.

y = –3x2 + 4x – 10

y = 5x2 – 9

y = x2 + 19

y = x2 + 5x – 2014

15

← upward→downward

2. The graph of a quadratic is a parabola, a u-shaped figure.

3. The parabola will open upward or downward.


16

A parabola that opens upward contains a vertex that is a minimum point. A parabola that opens downward contains a vertex that is a maximum point.

4.

vertex

vertex


17

5. The domain of a quadratic function is all real numbers.


18

6. To determine the range of a quadratic function, ask yourself two questions:

> Is the vertex a minimum or maximum? > What is the y-value of the vertex?

If the vertex is a minimum, then the range is all real numbers greater than or equal to the y-value.

The range of this quadratic is [–6,∞ )


19

If the vertex is a maximum, then the range is all real numbers less than or equal to the y-value.

The range of this quadratic is (–∞,10]


20

–b2ax =

An axis of symmetry (also known as a line of symmetry) will divide the parabola into mirror images. The line of symmetry is always a vertical line of the form

7.

x=2


y = 2x2 – 8x + 2 –(–8) 2(2)x = = 2 Teac

her N

otes

Remember "opposite of b." In this example, "b" is –8,

therefore the calculation is +8 divided by 4.

21

The x-intercepts are the points at which a parabola intersects the x-axis. These points are also known as zeroes, roots or solutions and solution sets. Each quadratic function will have two, one or no real x-intercepts.

8.


22

1 True or False: The vertex is the highest or lowest value on the parabola.

True

False

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2 If a parabola opens upward then...

A a>0

B a

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3 The vertical line that divides a parabola into two symmetrical halves is called...

A discriminantB perfect squareC axis of symmetryD vertex

E slice

25

4 What is the equation of the axis of symmetry of the

parabola shown in the diagram below?

A x=0.5

B x=2

C x=4.5

D x=15

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

26

5 The height, y, of a ball tossed into the air can be represented by the equation y = −x2 + 10x + 3, where x is the elapsed time. What is the equation of the axis of symmetry of this parabola?

A y=5B y=–5C x=5D x=–5


27

6 What are the vertex and axis of symmetry of the parabola shown in the diagram below?

A vertex: (1,−4); axis of symmetry: x = 1

B vertex: (1,−4); axis of symmetry: x = −4

C vertex: (−4,1); axis of symmetry: x = 1

D vertex: (−4,1); axis of symmetry: x = −4


28

7 What is the domain and range of the quadratic function?

A

B

C

D

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8 The equation y = x2 + 3x − 18 is graphed on the set of axes below. Based on this graph, what are the roots of the equation x2 + 3x − 18 = 0?

A –3 and 6

B 0 and –18

C 3 and –6

D 3 and –18


30

9 The equation y = − x2 − 2x + 8 is graphed on the set of axes below.

A 8 and 0

B 2 and –4

C 9 and –1

D 4 and –2

Based on this graph, what are the roots of the equation − x2 − 2x + 8 = 0?


31

10 What is the domain and range of the quadratic function?

A

B

C

D

32

Graphing Quadratic Functions in

Standard Form


33

Graph by Following Six Steps:

Step 1 - Find Axis of Symmetry

Step 2 - Find Vertex

Step 3 - Find Y intercept

Step 4 - Find two more points

Step 5 - Partially graph

Step 6 - Reflect

34

Axis of Symmetry

Step 1 - Find the Axis of Symmetry

What is the Axis of Symmetry?

Hint

The line that runs down the center of a parabola.

This line divides the graph into two perfect

halves.

35

Find the Axis of SymmetryGraph y = 3x2 – 6x + 1

Step 1

The axis of symmetryis x = 1.

Formula:a = 3b = -6

x = –b2a

x = –(–6) = 6 = 1 2(3) 6

36

y = 3(1)2 + –6(1) + 1y = 3 – 6 + 1y = –2Vertex = (1 , –2)

Find the vertex by substituting the value of x(the Axis of Symmetry) into the equation to get y.

Step 2

y = 3x2 – 6x + 1 a = 3, b = –6 and c = 1

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y-intercept

What is they-intercept?

Find y-intercept.Step 3

Hint

The point where the line passes through the

y-axis. This occurs when the x-value is 0.

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y = ax2 + bx + cy = 3x2 – 6x + 1

c = 1

The y-intercept is always the c value, because x = 0.

Step 3 - Find y-intercept.

Step 3Graph y = 3x2 – 6x + 1

The y-intercept is 1 and the graph passes through (0,1).

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Step 4Graph y = 3x2 – 6x + 1

Step 4 - Find two pointsChoose different values of x and plug

in to the equation find points.Let's pick x = –1 and x = –2

y = 3x2 – 6x + 1y = 3(–1)2 – 6(–1) + 1y = 3 + 6 + 1y = 10(–1,10)

y = 3x2 – 6x + 1y = 3(–2)2 – 6(–2) + 1y = 3(4) + 12 + 1y = 25(–2,25)

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Step 5

Step 5 - Graph the axis of symmetry, the vertex, the point containing the y-intercept and two other points.

41

(4,25)

Step 6Step 6 - Reflect the points across the axis of

symmetry. Connect the points with a smooth curve.

42

11 What is the axis of symmetry for y = x2 + 2x – 3 (Step 1)?

A 1

B –1

43

12 What is the vertex for y = x2 + 2x – 3 (Step 2)?

A (–1,–4)

B (1,–4)

C (–1,4)

44

13 What is the y-intercept for y = x2 + 2x – 3 (Step 3)?

A –3

B 3

45

14 What is an equation of the axis of symmetry of the parabola represented by y = −x2 + 6x − 4?

A x = 3

B y = 3

C x = 6

D y = 6


46

Graph y = 2x2 – 6x + 4

Graph

47

Graph f(x) = –x2 – 4x + 5

Graph

48

Graph y = 3x2 – 7

Graph

49

On the set of axes below, solve the following systemof equations graphically for all values of x and y.

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

y = –x2 – 4x + 12 y = –2x + 4

Solve Equations

50

On the set of axes below, solve the followingsystem of equations graphically for all values of x and y.

y = x2 − 6x + 1y + 2x = 6


Solve Equations

51

15 The graphs of the equations y = x2 + 4x – 1 and y + 3 = x are drawn on the same set of axes. At which point do the graphs intersect?

A (1,4)

B (1,–2)

C (–2,1)

D (–2,5)

From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011

52

Graphing Quadratic Functions in Vertex and Intercept Form


53

Standard Form of a Quadratic Equation is y = ax2 + bx + c.

Vertex Form of a Quadratic Equation is y = a (x – h)2 + k, where (h, k) is the vertex and x = h is the axis of symmetry.

Intercept Form of a Quadratic Equation is y = a (x – p) (x – q), where (p, 0) and (q, 0) are the roots (or zeros) of the graph.

Quadratic Equation Forms

54

Graph in Vertex Form by Following Five Steps:

Step 1 - Draw the axis of symmetry, x = h

Step 2 - Plot the vertex, (h, k)

Step 3 - Find two more points

Step 4 - Partially graph

Step 5 - Reflect

Graph in Vertex Form

55

Step 1 - Find the Axis of SymmetryGraph y = 2(x – 4) 2 – 3

y = 2(x – 4) – 3

The axis of symmetry is x = 4.

Step 1

56

Plot the VertexGraph y = 2(x – 4)2 – 3

y = 2(x – 4)2 – 3

The Vertex is (4, –3).

Step 2

57

Find two points

Graph y = 2(x – 4)2 – 3

Choose different values of x and plug in to the equation to find points.

y = 2(x – 4)2 – 3y = 2(2 – 4)2 – 3y = 2(2)2 – 3y = 8 (2, 5)

y = 2(x – 4)2 – 3y = 2(3 – 4)2 – 3y = 2(1)2 – 3y = –1(3, –1)

Let's pick x = 2 and x = 3

Step 3

58

Step 4Graph the axis of symmetry,

the vertex, and two other points.

59

Reflect the points across the axis of symmetry. Connect the points with a smooth curve.

Step 5

60

16 What is the axis of symmetry of the equationy = –(x + 5)2 + 1?

61

17 What is the vertex of y = –(x + 5)2 + 1?A (5,1)

B (1,5)

C (–5,1)

D (5,–1)

62

18 What is the axis of symmetry of the equation ofy = .5(x - 7) 2 + 3?

63

Graph y = –(x + 5)2 + 1

Graph y

64

f(x)

Graph f(x) = .5(x – 5)2 + 3

Graph f(x)

65

Graph in Intercept Form by Following Four Steps

Graph in Intercept Form

Step 1 - Draw the axis of symmetryStep 2 - Find and Plot the vertexStep 3 - Plot the rootsStep 4 - Graph

66

–2 + 42

x =

Step 1 - Find the Axis of Symmetry Graph y = –1/3(x + 2)(x – 4)

Use the formula

The axis of symmetry isx = 1.


x =p + q

2

67

Step 2 - Find and Plot the VertexGraph y = –1/3(x + 2)(x – 4)

y = –3 (x + 2) (x – 4)y = –3 (1 + 2) (1 – 4)y = –1/3 (3) (–3)y = 3


68

Step 3 - Find and Plot the RootsGraph y = –1/3(x + 2)(x – 4)

y = a(x – p)(x – q)y = –1/3(x + 2)(x – 4)

p = –2 and q = 4

(–2, 0) and (4, 0) are the roots


69

Step 4 - Connect the points with a smooth curve.


70

19 What is the axis of symmetry of the equationy = 4(x – 5) (x + 4)?

71

20 What is the vertex of the equation y = 4(x – 5)(x + 4)?

A (0.5,–77)

B (0.5,–81)

C (0.5,81)

D (–81,0.5)

72

21 What are the roots of the equation y = 4(x–5)(x+4)?

A (0–5),(0,4)

B (0,5),(0,–4)

C (–5,0),(4,0)

D (5,0),(–4,0)

73

Graph y = 2(x–3)(x+5)


74

Graph y = –.5(x–2)(x–8)


75

f(x)

Graph f(x) = (x + 1)2 – 4


76

f(x)

Graph f(x) = –0.5(x + 3)(x – 3)


77

Transforming and TranslatingQuadratic Functions


78

x x2

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

The quadratic parent function is f(x) = x2. The graph of all other quadratic functions are transformations of the graph of f(x) = x2.

Quadratic Functions

y = x2

79

The quadratic parent function is f(x) = x 2.How is f(x) = x2 into f(x) = 2x 2?

x 2

-3 18

-2 8

-1 2

0 0

1 2

2 8

3 18

Quadratic Functions

y = x2

y = 2x2

80

x 0.5

-3 4.5

-2 2

-1 0.5

0 0

1 0.5

2 2

3 4.5

The quadratic parent function is f(x) = x 2. How is f(x) = x2 into f(x) = .5x 2?

Quadratic Functions

y = x2

y = x212

81

What does "a" do in y = ax2 + bx + c?

How does a>0 effect the parabola? How does a

82

What does "a" do in y = ax2 + bx + c?How does your conclusion about "a" change as a changes?

y = x2y = x212

y = 3x2

y = –1x2 y = –3x2 y = – x212

83

If the absolute value of a is > 1, then the graph of the function is narrower than the graph of the parent function.

If the absolute value of a is < 1, then the graph of the function is wider than the graph of the parent function.

If a > 0, the graph opens up.

If a < 0, the graph opens down.

What does "a" do in y = ax2 + bx + c?

84

0 1

parent function

narrowerwider

2

Consider an "absolute value" number line to compare a parabola to parent function.

Absolute Value

y = 0.5x2

y = x2

y = 1.75x2

85

22 Without graphing determine which direction does the parabola open and if the graph is wider or narrower than the parent function.

A up, wider

B up, narrower

C down, wider

D down, narrower

y = .3x2

86


A up, wider

B up, narrower

C down, wider

D down, narrower

y = –4x2

87


A up, wider

B up, narrower

C down, wider

D down, narrower

y = –4x2 + 100x + 45

88

y = – x223


A up, wider

B up, narrower

C down, wider

D down, narrower

89


A up, wider

B up, narrower

C down, wider

D down, narrower

y = – x275

90

What does "c" do in y = ax2 + bx + c?

y = x2 + 6

y = x2 + 3

y = x2

y = x2 – 2

y = x2 – 5y = x2 – 9

91

What does "c" do in y = ax2 + bx + c?

"c" moves the graph up or down the same value as "c."

"c" is the y-intercept, when the graph is in standard form.

92

27 Without graphing, what is the y-intercept ofthe given equation?

y = x2 + 17

93

28 Without graphing, what is the y-interceptof the given equation?

y = –x2 – 6

94

29 Without graphing, what is the y-interceptof the given function?

f(x) = –3x2 + 13x – 9

95

30 Without graphing, what is the y-interceptof the given equation?

y = 2x2 + 5x

96

31 Choose all that apply to the following quadratic:

A opens upB opens down

C wider than parent function

D narrower than parent function

E y-intercept of y = –4

F y-intercept of y = –2

G y-intercept of y = 0

H y-intercept of y = 2I y-intercept of y = 4

J y-intercept of y = 6

f(x) = –.7x2 – 4

97

32 Choose all that apply to the following quadratic:

A opens upB opens down

C wider than parent function

D narrower than parent function

E y-intercept of y = –4

F y-intercept of y = –2

G y-intercept of y = 0

H y-intercept of y = 2I y-intercept of y = 4

J y-intercept of y = 6

f(x) = x2 – 6x43

98

33 The diagram below shows the graph of y = –x2 – c.

A B C D

Which diagram shows the graph of y = x2 – c?


99

What does "d" do in y = (x – d)2?

y = (x + 4)2 y = x2 y = (x - 4)2

100

"d" moves the graph left or right, the same value as "d."

In this form "d" is the x-intercept.

What does "d" do in y = (x – d)2?

101

34 Without graphing, what is the x-intercept of the given equation?

A (2, 0)

B (–2,0)

y = (x + 2)2

102

35 Without graphing, what is the x-interceptof the given function?

A (7,0)

B (–7,0)

f(x) = (x – 7)2

103

36 Without graphing, what is the x-interceptof the given function?

f(x) = (x + 11)2

104

37 For the following quadratic, choose all that apply...

A opens up

B opens down

C wider than the parent function

D narrower than the parent function

E x-intercept is (6,0)

F x-intercept is (–6,0)

G x-intercept is (–3,0)

H y-intercept is (0,6)

I y-intercept is (0,–6)

J y-intercept is (0,–3)

f(x) = –5x2 – 3x + 6

105

38 For the following quadratic, choose all that apply...

A opens up

B opens down

C wider than the parent function


E x-intercept is (6,0)

F x-intercept is (-6,0)G x-intercept is (-3,0)

H y-intercept is (0,6)

I y-intercept is (0,-6)

J y-intercept is (0.-3)

f(x) = (x + 6)214

106

39 For the following question, choose all that apply...

A opens up

B opens down

C wider thant the parent function


E vertex is (–5,9)

F vertex is (5,–9)

G vertex is (–5,–9)

H vertex is (–2,5)

f(x) = –2(x + 5)2 + 9

107


Comparison of Types of Functions

108

Thus far we have learned:

1. The characteristics of a quadratic function. 2. How to graph a quadratic function. 3. Transformations and translations of quadratic

functions.

Next we will compare at least two different functions to each other. We will look at Quadratic, Linear, and Exponential Functions in terms of y-intercept, rate of change at an interval, maximum or minimum at an interval, and evaluate at a point.

Review

109

f(x)

x

x g(x)

–6 –2

–3 –0.5

0 1

4 3

In the next few questions we will compare key features of the two functions below. The linear function represented by the table and the quadratic function represented by the graph.

Comparison Functions

110

f(x)

x

x g(x)

–6 –2

–3 –0.5

0 1

4 3

Compare the y-intercepts of the functions.

The y-intercept of g(x) is (0,1) and f(x) is (0,–1)

g(x) > f(x)

Compare y-intercepts

111

f(x)

x

x g(x)

–6 –2

–3 –0.5

0 1

4 3

Compare the functions at f(2).

Referring to the graph for f(x), f(2) is approximately 3.Referring to the table of values for g(x) is between 1 and 3.

At f(2) g(x) < f(x)

Compare Functions

112

f(x)

x g(x)

–6 –2

–3 –0.5

0 2

4 3

Compare the rate of change over the interval –3 ≤ x ≤ 0

To find the rate of change of the functions,find the slope at the intervals.

Compare Rate of Change

x

113

At the interval g(x) > f(x)

= = =∆g(x)∆x

g(0) – g(–3)0 – (–3)

1 – (–0.5)0 + 3

12

= = = –1∆f(x)∆x

f(0) – f(–3)0 – (–3)

–1 –20 + 3

Compare Rate of ChangeCompare the rate of change over the interval –3 ≤ x ≤ 0

slope(m) = =∆y∆x

y2 – y1x2 – y1

114

40 Compare the y-intercepts of the functions.

A g(x)h(x)

C g(x)=h(x)

g(x)

x

x –2 0 2 5f(x) –13 –7 –1 3

115


A g(x)h(x)

C g(x)=h(x)

g(x)

x

x –2 0 2 5f(x) –13 –7 –1 3

116

42 Compare the rate of change of the functions over the interval –2≤ x ≤ 2.

A g(x)h(x)

C g(x)=h(x)

g(x)

x

x –2 0 2 5f(x) –13 –7 –1 3

117


A g(x)h(x)

C g(x)=h(x)

g(x)x

x –5 –1 1 3h(x) –13 –5 –1 3

118

44 Compare g(2) and h(2).

A g(x)h(x)

C g(x)=h(x)

g(x)x

x –5 –1 1 3h(x) –13 –5 –1 3

119

45 Compare the minimum value g(x) to f(x).

A g(x)f(x)

C g(x)=f(x)

g(x)

x

f(x) = (x – 3)(x + 4)14

Hin

t

120

46 Compare the maximum value of h(x) to f(x).

A h(x)f(x)

C h(x)=f(x)

h(x) = –7(x – 12)2 + 41

f(x) = –14x2 + 23x – 57

Hin

t

121

47 Compare the rate of change of the functionson the interval

A g(x)f(x)

C g(x)=f(x)

x –2 0 2 5f(x) –3 –1 1 4

122

In the next few questions we will compare key features of the two functions below. The linear function represented by the graph and the exponential function represented by the table.

x h(x)

0 1

1 3

2 9

3 27

Compare Functions

g(x)

x

123

x h(x)

0 1

1 3

2 9

3 27

Compare the y-intercepts of the functions.

The y-intercept of g(x) is (0,1). The y-intercept of h(x) is (0,1). The y-intercepts of the functions are equal.

g(x)

x

Compare y-intercepts

124

x h(x)

0 1

1 3

2 9

3 27

g(x)

Compare the rate of change over the interval 0 ≤ x ≤ 1.


125



At the interval g(x) > h(x)

= = =∆g(x)∆x

g(1) – g(0)1 – 0

4 – 11

3

= = = 2∆h(x)∆x

f(1) – f(0)1 – 0

3 – 11

126

x h(x)

0 1

1 3

2 9

3 27

g(x)

x



127



At the interval g(x) < h(x)

= = =∆g(x)∆x

g(2) – g(1)2 – 1

7 – 41

3

= = = 6∆h(x)∆x

h(2) – h(1)2 – 1

9 – 31

128


A g(x)f(x)

x –1 0 1 2f(x) 1.5 3 6 12

g(x)x

129

49 Compare the rate of change over the interval -2 ≤ x ≤ 0

A g(x)f(x)

x -2 0 2 4

f(x) 0.75 3 12 48

g(x)

x

130

50 Compare the rate of change over the interval 0 ≤ x ≤ 2

A g(x)f(x)

x -2 0 2 4

f(x) 0.75 3 12 48

g(x)x

131


A h(x)f(x)

f(x) = 1/5x +6

h(x)

x

132

52 Compare the rate of change over the interval 0 ≤ x ≤ 1

A h(x)f(x)

f(x) = 1/5x +6

h(x)

x

133

The next four problems are comparisons between linear, quadratic, and exponential functions.

Next Four Problems

134

g(x)

h(x)

f(x)


A g(x)

135

g(x)

h(x)

f(x)

54 Which function has the greatest rate of changeover the given interval?

A g(x) B h(x) C f(x)

–2 ≤ x ≤ –1

Answ

er

∆h(x)∆x

∆g(x)∆x

∆f(x)∆x

136



–1 ≤ x ≤ –1

g(x)

h(x)

f(x)

Answ

er

∆h(x)∆x

∆g(x)∆x

∆f(x)∆x

137



0 ≤ x ≤ 1

g(x)

h(x)

f(x)

Answ

er

Documents

Algebra I - NJCTLcontent.njctl.org/.../non-linear-functions-2-2015-11-04.pdf2015/11/04 · ax2 + bx + c = 0 , where a is not equal to 0. The form ax2 + bx + c = 0 is called the standard