quadratic transformations assignment

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2 Transformations of Quadratic Functions 107

Work with a partner. In parts (a)–(f), match each quadratic function with its graph. Explain your reasoning. Then use technology to check your answer. a. g(x) = −(x − 2) 2 b. g(x) = (x − 2) 2 + 2 c. g(x) = −(x + 2) 2 − 2

####### EXPLORE ITEXPLORE IT Identifying Graphs of Quadratic Functions

Transformations of

Quadratic functions.

Learning Target:

Success Criteria:

Describe and graph transformations of quadratic functions.

• I can describe transformations of quadratic functions.
• I can graph transformations of quadratic functions.
• I can write functions that represent transformations of quadratic functions.

Functions MA.912.F.2 Identify the effect on the graph of a given function of two or more transformations defi ned by adding a real number to the x- or y-values or multiplying the x- or y-values by a real number. MA.912.F.2 Given a table, equation or graph that represents a function, create a corresponding table, equation or graph of the transformed function defined by adding a real number to the x- or y-values or multiplying the x- or y-values by a real number. Also MA.912.AR.3, MA.912.F.

d. g(x) = 0(x − 2) 2 − 2 e. g(x) = 2(x − 2) 2 f. g(x) = −(x + 2) 2 + 2

–4 –2 2 4 -

g. How do the constants a, h, and k affect the graph of the quadratic function g(x) = a(x − h) 2 + k?

MAKE A CONNECTION How can your knowledge of transformations of absolute value functions help you match each function with its graph?

© Big Ideas Learning, LLC

108 Chapter 2 Quadratic Functions

####### Describing Transformations of Quadratic Functions

A quadratic function is a function that can be written in the form f(x) = a(x − h) 2 + k, where a ≠ 0. The U-shaped graph of a quadratic function is called a parabola. In Section 1, you graphed quadratic functions using tables of values. You can also graph quadratic functions by applying transformations to the graph of the parent function f(x) = x2.

Vocabulary quadratic function, p. 108 parabola, p. 108 vertex of a parabola, p. 112 vertex form, p. 112

Horizontal Translations f (x − h) = (x − h) 2

y y = (x − h)2, h < 0

y = (x − h)2, h > 0

● shifts left when h < 0 ● shifts right when h > 0

Reflections in the x-Axis −f(x) = −(x2) = −x 2

y = – x 2 fl ips over the x-axis

Horizontal Stretches and Shrinks f(ax) = (ax) 2

y = (ax)2, 0 < a < 1

y = (ax) 2, a > 1

● horizontal stretch (away from y-axis) by a factor of 1 — a

when 0 < a < 1

● horizontal shrink (toward y-axis) by a factor of 1 — a

when a > 1

Vertical Translations f (x) + k = x 2 + k

y = x 2 + k, k < 0

y = x 2 + k, k > 0

● shifts down when k < 0 ● shifts up when k > 0

Reflections in the y-Axis f (−x) = (−x) 2 = x 2

y = x 2 is its own reflection in the y-axis.

Vertical Stretches and Shrinks

a ⋅ f (x) = ax 2

y = ax 2 , 0 < a < 1

y = ax2, a > 1

● vertical stretch (away from x-axis) by a factor of a when a > 1 ● vertical shrink (toward x-axis) by a factor of a when 0 < a < 1

Let f (x) = x2.

110 Chapter 2 Quadratic Functions

SELF-ASSESSMENT 1 I don’t understand yet. 2 I can do it with help. 3 I can do it on my own. 4 I can teach someone else.

Describe the transformation of f(x) = x 2 represented by g. Then graph each function.

• g(x) = (x − 3) 2 2. g(x) = (x − 2) 2 − 2 3. g(x) = (x + 5) 2 + 1

4. g(x) = ( 1 — 3 x )

2 5. g(x) = 3(x − 1) 2 6. g(x) = −(x + 3) 2 + 2

• STRUCTURE The graph of g(x) = 9 x 2 is a vertical stretch by a factor of 9 of the graph of f(x) = x2. Describe the graph of g as a horizontal shrink of the graph of f. Explain your reasoning.

EXAMPLE 3 Describing a Transformation

Describe the transformation from the graph of f to the graph of g.

The function g(x) = f (x − h) indicates that the graph of g is a horizontal translation of the graph of f. The graphs of f and g show that for any input, g(x) = f (x + 3). For example, g(0) = 1 and f (3) = 1. So, g(x) = f (x + 3) and h = −3.

The graph of g is a translation 3 units left of the graph of f.

− 4 − 2 2 4 x

− 44 − 22 g(x) = f(x + 3)

f(x) = x 2 − 4 x + 4

− 44 − 22 g(x) = f(x − h)

2 Transformations of Quadratic Functions 111

EXAMPLE 4 Describing a Combination of Transformations

Describe the transformation of f (x) = − 2 x 2 + x represented by g(x) = f (0) + 1. Then graph each function.

Notice that the function is of the form g(x) = f (ax) + k, where a = 0 and k = 1.

So, the graph of g is a horizontal stretch by a factor of —0 1 = 2 and a vertical translation 1 unit up of the graph of f.

Describe the transformation from the graph of f to the graph of g. 8.

− 4 − 2 2 4

f(x) = −x 2 − 3

g(x) = f(x) + k

f(x) = x 2 − 4 x

g(x) = a ∙ f(x)

• The table represents two quadratic functions f and g. Describe the transformation from the graph of f to the graph of g.

Describe the transformation of f represented by g. Then graph each function.

11. f (x) = x 2 + x + 1; g(x) = — 12 ⋅ f (x + 4) 12. f(x) = − — 13 x 2 + 2; g(x) = f (−x) − 1

x − 1 0 1 2 3 f (x) 7 1 − 1 1 7 g (x) 8 2 0 2 8

2 Transformations of Quadratic Functions 113

• Let the graph of g be a vertical shrink by a factor of 1 — 2 , followed by a translation 2 units up of the graph of f (x) = x2. Write a rule for g and identify the vertex.

EXAMPLE 6 Writing a Transformed Quadratic Function

Let the graph of g be a translation 3 units right and 2 units up, followed by a refl ection in the y-axis of the graph of f(x) = x 2 − 5 x. Write a rule for g.

SOLUTION Step 1 First write a function h that represents the translation of f. h(x) = f(x − 3) + 2 Subtract 3 from the input. Add 2 to the output. = (x − 3)2 − 5(x − 3) + 2 Replace x with x − 3 in f(x). = x 2 − 11 x + 26 Simplify. Step 2 Then write a function g that represents the reflection of h. g(x) = h(−x) Multiply the input by −1. = (−x) 2 − 11(−x) + 26 Replace x with −x in h(x). = x 2 + 11 x + 26 Simplify.

REMEMBER To multiply two binomials, use the FOIL Method. First Inner (x + 1)(x + 2) = x 2 + 2 x + x + 2 Outer Last

114 Chapter 2 Quadratic Functions

EXAMPLE 7 Modeling Real Life

The height (in feet) of water spraying from a fire hose can be modeled by h(x) = −0 2 + x + 25, where x is the horizontal distance (in feet) from the fire truck. The crew raises the aerial ladder so that the water hits the ground 10 feet farther from the fi re truck. Write a function that models the new path of the water.

SOLUTION 1. Understand the Problem You are given a function that represents the path of water spraying from a fire hose. You are asked to write a function that represents the path of the water after the crew raises the ladder. 2. Make a Plan Analyze the graph of the function to determine the translation of the ladder that causes water to travel 10 feet farther. Then write the function. 3. Solve and Check Use technology to graph the original function. Because h(50) = 0, the water originally hits the ground 50 feet from the fire truck. After raising the ladder, h(60) = 0. So, by observing that h(60) = −23 before the crew raises the ladder, you can determine that a translation 23 feet up causes the water to travel 10 feet farther from the fire truck. g(x) = h(x) + 23 Add 23 to the output. = −0 2 + x + 48 Substitute for h(x) and simplify.

The new path of the water can be modeled by g(x) = −0 2 + x + 48.

Check Verify that g(60) = 0.

g(60) = −0(60) 2 + 60 + 48 = − 108 + 60 + 48 = 0 ✓

y = −0 2 + x + 25

(50, 0) 20 40 60

Let the graph of g be a translation 4 units left, followed by a horizontal shrink by a factor of — 13 of the graph of f (x) = x 2 + x. Write a rule for g.

WHAT IF? In Example 7, the water hits the ground 10 feet closer to the fire truck after lowering the ladder. Write a function that models the new path of the water.

116 Chapter 2 Quadratic Functions

In Exercises 15–22, describe the transformation of f(x) = x 2 represented by g. Then graph each function. (See Example 2.)

g(x) = −x 2 16. g(x) = (−x) 2 17. g(x) = 3 x 2

g(x) = — 13 x 2 19. g(x) = (2x) 2 20. g(x) = −(2x) 2

g(x) = — 15 x 2 − 4 22. g(x) = — 12 (x − 1) 2

REASONING In Exercises 11–14, match the function with the correct transformation of the graph of f. Explain your reasoning.

y = f(x − 1) 12. y = f(x) + 1

y = f(x − 1) + 1 14. y = f(x + 1) − 1

2 Transformations of Quadratic Functions 117

In Exercises 23–30, describe the transformation from the graph of f to the graph of g. (See Example 3.)

x 0 1 2 3 f (x) 5 6 9 14 g(x) = f (x) + k 7 8 11 16

x − 2 − 1 0 1 f (x) 4 1 0 1

g(x) = a ⋅ f (x) − 4 − 1 0 − 1

22 x g(x) = a ∙ f(x)

g(x) = f(x − h)

x − 4 − 2 0 2 f (x) 39 15 7 15 g(x) 36 12 4 12

x − 3 − 1 1 3 f (x) − 5 3 3 − 5 g(x) 5 − 3 − 3 5

In Exercises 31–34, describe the transformation of f represented by g. Then graph each function. (See Example 4.)

f (x) = x 2 + 4; g(x) = f (−x) + 5 32. f (x) = 3 x 2 − 18; g(x) = 1 — 6 f (x − 2)

f (x) = x 2 − 2 x + 5; g(x) = −f (4x) 34. f (x) = −0 2 + x − 1; g(x) = − 3 f (x) + 3

2 Transformations of Quadratic Functions 119

STRUCTURE In Exercises 47–52, match the function with its graph. Explain your reasoning. 47. g(x) = 2(x − 1) 2 − 2 48. g(x) = — 12 (x + 1) 2 − 2 49. g(x) = −2(x − 1) 2 + 2

• g(x) = 2(x + 1) 2 + 2 51. g(x) = −2(x + 1) 2 − 2 52. g(x) = 2(x − 1) 2 + 2

JUSTIFYING STEPS In Exercises 53 and 54, justify each step in writing a rule for g described by the transformations of f(x) = 2 x 2 + 6 x. 53. translation 6 units down, followed by a refl ection in the x-axis h(x) = f(x) − 6 = 2 x 2 + 6 x − 6

g(x) = −h(x) = −(2x 2 + 6 x − 6) = − 2 x 2 − 6 x + 6

• refl ection in the y-axis, followed by a translation 4 units right h(x) = f(−x) = 2(−x) 2 + 6(−x) = 2 x 2 − 6 x

g(x) = h(x − 4) = 2(x − 4) 2 − 6(x − 4) = 2 x 2 − 22 x + 56

120 Chapter 2 Quadratic Functions

MODELING REAL LIFE A grasshopper can jump incredible distances, up to 20 times its length. The height (in inches) of a 1-inch-long grasshopper during a jump can be modeled by h(x) = − — 201 x 2 + x, where x is the horizontal distance traveled (in inches). When the grasshopper jumps off a rock, it lands on the ground 2 inches farther away. Write a function that models the new path of the jump. (See Example 7.)

MODELING REAL LIFE The height (in feet) of a red kangaroo during a jump can be modeled by h(x) = −0(x − 14) 2 + 6, where x is the horizontal distance traveled (in feet). When the kangaroo jumps from a higher location, it lands on the ground 5 feet farther away. Write a function that models the new path of the jump.

s from a higher round 5 feet tion that models

PROBLEM SOLVING The function f(t) = − 16 t 2 + 10 models the height (in feet) of an object t seconds after it is dropped from a height of 10 feet on Earth. The same object dropped from the same height on the moon is modeled by g(t) = − 8 — 3 t 2 + 10. Describe the transformation of the graph of f to obtain g. From what height must the object be dropped on the moon so it hits the ground after the same number of seconds as on Earth?

HOW DO YOU SEE IT? Describe the transformation of f(x) = x 2 represented by g.

• PROBLEM SOLVING The graph of h is a horizontal shrink by a factor of 1 — 4 and a translation 1 unit up and 3 units to the left of the graph of g(x) = 4 x 2 − 4. Describe the transformations of the graph of the parent quadratic function represented by g and h.

122 Chapter 2 Quadratic Functions

REVIEW & REFRESHREVIEW & REFRESH

Find the measure of each angle in the isosceles trapezoid.

Let the graph of g be a translation 1 unit left and 6 units down, followed by a vertical shrink by a factor of — 12 of the graph of f(x) = 3(x + 2) 2. Write a rule for g and identify the vertex.

In Exercises 67 and 68, write an inequality that represents the graph.

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

0 1 2 3 4 5

Factor 8x 4 − 40 x3.

Use intercepts to graph the equation 4x − 5 y = 30.

Write a piecewise function represented by the graph.

Describe the transformation of f(x) = x 2 represented by g.

In Exercises 73–76, solve the inequality. Graph the solution, if possible.

∣ k ∣ > 2

∣ f − 4 ∣ ≤ − 3

∣ m − 7 ∣ < 72

∣ 4 f + 9 ∣ ≥ 81

MODELING REAL LIFE The cost (in dollars) of x pairs of jeans at a store’s regular price is modeled by the function f(x) = 40 x. The cost of x pairs of jeans during a sale is $5 less than 85% of the cost at regular price. What is the cost of three pairs of jeans during the sale?

In Exercises 78 and 79, graph f. Determine when the function is positive, negative, increasing, or decreasing. Then describe the end behavior of the function.

f (x) = ∣ x + 2 ∣ − 5 79. f (x) = − 4 ∣ x − 1 ∣ + 3

Find the perimeter of the polygon with the vertices A(−1, −3), B(−1, 4), C(2, 2), and D(2, −3).

• Multiple Choice

Subject : Algebra 2

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