Section 2.1 Transformations of Quadratic Functions 47
Essential Question
Essential Question How do the constants a, h, and k affect the
graph of the quadratic function g(x) = a(x − h)
2
+ k?
The parent function of the quadratic family is f(x) = x
2
. A transformation of the
graph of the parent function is represented by the function g(x) = a(x − h)
2
+ k,
where a ≠ 0.
Learning Standards
HSF-IF.C.7c
HSF-BF.B.3
COMMON
CORE
Identifying Graphs of Quadratic Functions
Work with a partner. Match each quadratic function with its graph. Explain your
reasoning. Then use a graphing calculator to verify that your answer is correct.
a. g(x) = −(x − 2)
2
b. g(x) = (x − 2)
2
+ 2 c. g(x) = −(x + 2)
2
− 2
d. g(x) = 0.5(x − 2)
2
− 2 e. g(x) = 2(x − 2)
2
f. g(x) = −(x + 2)
2
+ 2
A.
6
−4
−6
4
B.
6
−4
−6
4
C.
6
−4
−6
4
D.
6
−4
−6
4
E.
6
−4
−6
4
F.
6
−4
−6
4
Communicate Your Answer
Communicate Your Answer
2. How do the constants a, h, and k affect the graph of
the quadratic function g(x) = a(x − h)
2
+ k?
3. Write the equation of the quadratic function whose
graph is shown at the right. Explain your reasoning.
Then use a graphing calculator to verify that your
equation is correct.
LOOKING FOR
STRUCTURE
To be pro cient in math,
you need to look closely
to discern a pattern or
structure.
6
−4
−6
4
2.1
Transformations of
Quadratic Functions
48 Chapter 2 Quadratic Functions
2.1
Lesson
What You Will Learn
What You Will Learn
Describe transformations of quadratic functions.
Write transformations of 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 1.1, 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) = x
2
.
quadratic function, p. 48
parabola, p. 48
vertex of a parabola, p. 50
vertex form, p. 50
Previous
transformations
Core Vocabulary
Core Vocabu
l
l
a
r
r
y
Core
Core
Concept
Concept
Horizontal Translations
f(x) = x
2
f(x − h) = (x − h)
2
y
y = (x − h)
2
,
h < 0
y = (x − h)
2
,
h > 0
y = x
2
x
●
shifts left when h < 0
●
shifts right when h > 0
Vertical Translations
f(x) = x
2
f(x) + k = x
2
+ k
y
y = x
2
+ k,
k < 0
y = x
2
+ k,
k > 0
y = x
2
x
●
shifts down when k < 0
●
shifts up when k > 0
Translations of a Quadratic Function
Describe the transformation of f(x) = x
2
represented by g(x) = (x + 4)
2
− 1. Then
graph each function.
SOLUTION
Notice that the function is of the form
x
y
4
6
2
−2 2−6
g
f
g(x) = (x − h)
2
+ k. Rewrite the function
to identify h and k.
g(x) = (x − (−4))
2
+ (−1)
h k
Because h = −4 and k = −1, the graph
of g is a translation 4 units left and 1 unit
down of the graph of f.
Monitoring Progress
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Describe the transformation of f(x) x
2
represented by g. Then graph
each function.
1. g(x) = (x − 3)
2
2. g(x) = (x − 2)
2
− 2 3. g(x) = (x + 5)
2
+ 1
Section 2.1 Transformations of Quadratic Functions 49
Core
Core
Concept
Concept
Re ections in the
x
-Axis
f(x) = x
2
−f(x) = −(x
2
) = −x
2
x
y
y = x
2
y = – x
2
ips over the x-axis
Horizontal Stretches and Shrinks
f(x) = x
2
f(ax) = (ax)
2
x
y
y = x
2
y = (ax)
2
,
0 < a < 1
y = (ax)
2
,
a > 1
●
horizontal stretch (away from
y-axis) when 0 < a < 1
●
horizontal shrink (toward y-axis)
when a > 1
Re ections in the
y
-Axis
f(x) = x
2
f(−x) = (−x)
2
= x
2
x
y
y = x
2
y = x
2
is its own re ection
in the y-axis.
Vertical Stretches and Shrinks
f(x) = x
2
a
⋅
f(x) = ax
2
x
y
y = x
2
y = ax
2
,
0 < a < 1
y = ax
2
,
a > 1
●
vertical stretch (away from
x-axis) when a > 1
●
vertical shrink (toward x-axis)
when 0 < a < 1
Transformations of Quadratic Functions
Describe the transformation of f(x) = x
2
represented by g. Then graph each function.
a. g(x) = −
1
—
2
x
2
b. g(x) = (2x)
2
+ 1
SOLUTION
a. Notice that the function is of the form
g(x) = −ax
2
, where a =
1
—
2
.
So, the graph of g is a re ection
in the x-axis and a vertical shrink
by a factor of
1
—
2
of the graph of f.
x
y
f
g
2
−2
2−2
b. Notice that the function is of the
form g(x) = (ax)
2
+ k, where a = 2
and k = 1.
So, the graph of g is a horizontal
shrink by a factor of
1
—
2
followed
by a translation 1 unit up of the
graph of f.
x
y
f
g
4
6
2−2
LOOKING FOR
STRUCTURE
In Example 2b, notice that
g(x) = 4x
2
+ 1. So, you can
also describe the graph of
g as a vertical stretch by
a factor of 4 followed by
a translation 1 unit up of
the graph of f.
50 Chapter 2 Quadratic Functions
Monitoring Progress
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Describe the transformation of f(x) x
2
represented by g. Then graph
each function.
4. g(x) =
(
1
—
3
x
)
2
5. g(x) = 3(x − 1)
2
6. g(x) = −(x + 3)
2
+ 2
Writing Transformations of Quadratic Functions
The lowest point on a parabola that opens up or the highest point on a parabola
that opens down is the vertex. The vertex form of a quadratic function is
f(x) = a(x − h)
2
+ k, where a ≠ 0 and the vertex is (h, k).
f(x) = a(x − h)
2
+ k
k
indicates a
vertical translation.
a
indicates a reflection
in the
x
-axis and/or a
vertical stretch or shrink.
h
indicates
a horizontal
translation.
Writing a Transformed Quadratic Function
Let the graph of g be a vertical stretch by a factor of 2 and a re ection in the x-axis,
followed by a translation 3 units down of the graph of f(x) = x
2
. Write a rule for g and
identify the vertex.
SOLUTION
Method 1 Identify how the transformations affect the constants in vertex form.
re ection in x-axis
a = −2
vertical stretch by 2
translation 3 units down} k = −3
Write the transformed function.
g(x) = a(x − h)
2
+ k Vertex form of a quadratic function
= −2(x − 0)
2
+ (−3) Substitute −2 for
a
, 0 for
h
, and −3 for
k
.
= −2x
2
− 3 Simplify.
The transformed function is g(x) = −2x
2
− 3. The vertex is (0, −3).
Method 2 Begin with the parent function and apply the transformations one at a time
in the stated order.
First write a function h that represents the re ection and vertical stretch
of f.
h(x) = −2
⋅
f(x) Multiply the output by −2.
= −2x
2
Substitute
x
2
for
f
(
x
).
Then write a function g that represents the translation of h.
g(x) = h(x) − 3 Subtract 3 from the output.
= −2x
2
− 3 Substitute −2
x
2
for
h
(
x
).
The transformed function is g(x) = −2x
2
− 3. The vertex is (0, −3).
Check
5
−20
−5
20
g
f
Section 2.1 Transformations of Quadratic Functions 51
Writing a Transformed Quadratic Function
Let the graph of g be a translation 3 units right and 2 units up, followed by a re ection
in the y-axis of the graph of f(x) = x
2
− 5x. 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
− 11x + 26 Simplify.
Step 2 Then write a function g that represents the re ection 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
+ 11x + 26 Simplify.
Modeling with Mathematics
The height h (in feet) of water spraying from a re hose can be modeled by
h(x) = −0.03x
2
+ x + 25, where x is the horizontal distance (in feet) from the re
truck. The crew raises the ladder so that the water hits the ground 10 feet farther from
the 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 re 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 the Problem Graph the transformed function.
Because h(50) = 0, the water originally hits the ground 50 feet from the re
truck. The range of the function in this context does not include negative values.
However, by observing that h(60) = −23, you can determine that a translation
23 units (feet) up causes the water to travel 10 feet farther from the re truck.
g(x) = h(x) + 23 Add 23 to the output.
= −0.03x
2
+ x + 48 Substitute for
h
(
x
) and simplify.
The new path of the water can be modeled by g(x) = −0.03x
2
+ x + 48.
4. Look Back To check that your solution is correct, verify that g(60) = 0.
g(60) = −0.03(60)
2
+ 60 + 48 = −108 + 60 + 48 = 0 ✓
Monitoring ProgressMonitoring Progress
Help in English and Spanish at BigIdeasMath.com
7. 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) = x
2
. Write a rule for g and identify the vertex.
8. Let the graph of g be a translation 4 units left followed by a horizontal shrink by a
factor of
1
—
3
of the graph of f(x) = x
2
+ x. Write a rule for g.
9. WHAT IF? In Example 5, the water hits the ground 10 feet closer to the re truck
after lowering the ladder. Write a function that models the new path of the water.
REMEMBER
To multiply two binomials,
use the FOIL Method.
First Inner
(x + 1)(x + 2) = x
2
+ 2x + x + 2
Outer Last
X=50 Y=0
80
−30
0
60
y = −0.03x
2
+ x + 25
T
h
t
t
S
HSCC_Alg2_PE_02.1.indd 51HSCC_Alg2_PE_02.1.indd 51 5/28/14 11:48 AM5/28/14 11:48 AM
52 Chapter 2 Quadratic Functions
Exercises
2.1
Dynamic Solutions available at BigIdeasMath.com
Monitoring Progress and Modeling with MathematicsMonitoring Progress and Modeling with Mathematics
In Exercises 3–12, describe the transformation of
f(x) x
2
represented by g. Then graph each function.
(See Example 1.)
3. g(x) = x
2
− 3 4. g(x) = x
2
+ 1
5. g(x) = (x + 2)
2
6. g(x) = (x − 4)
2
7. g(x) = (x − 1)
2
8. g(x) = (x + 3)
2
9. g(x) = (x + 6)
2
− 2 10. g(x) = (x − 9)
2
+ 5
11. g(x) = (x − 7)
2
+ 1 12. g(x) = (x + 10)
2
− 3
ANALYZING RELATIONSHIPS
In Exercises 13–16, match the
function with the correct
transformation of the graph
of f. Explain your reasoning.
13. y = f(x − 1) 14. y = f(x) + 1
15. y = f(x − 1) + 1 16. y = f(x + 1) − 1
A.
x
y
B.
x
y
C.
x
y
D.
x
y
In Exercises 17–24, describe the transformation of
f(x) x
2
represented by g. Then graph each function.
(See Example 2.)
17. g(x) = −x
2
18. g(x) = (−x)
2
19. g(x) = 3x
2
20. g(x) =
1
—
3
x
2
21. g(x) = (2x)
2
22. g(x) = −(2x)
2
23. g(x) =
1
—
5
x
2
− 4 24. g(x) =
1
—
2
( x − 1)
2
ERROR ANALYSIS In Exercises 25 and 26, describe
and correct the error in analyzing the graph of
f(x) 6x
2
4.
25.
The graph is a re ection in the
y-axis and a vertical stretch
by a factor of 6, followed by a
translation 4 units up of the graph
of the parent quadratic function.
✗
26.
The graph is a translation 4 units
down, followed by a vertical
stretch by a factor of 6 and a
re ection in the x-axis of the graph
of the parent quadratic function.
✗
USING STRUCTURE In Exercises 27–30, describe the
transformation of the graph of the parent quadratic
function. Then identify the vertex.
27. f(x) = 3(x + 2)
2
+ 1
28. f(x) = −4(x + 1)
2
− 5
29. f(x) = −2x
2
+ 5
30. f(x) =
1
—
2
( x − 1)
2
1. COMPLETE THE SENTENCE The graph of a quadratic function is called a(n) ________.
2. VOCABULARY Identify the vertex of the parabola given by f(x) = (x + 2)
2
− 4.
Vocabulary and Core Concept Check
p
p
p
p
x
y
f
HSCC_Alg2_PE_02.1.indd 52HSCC_Alg2_PE_02.1.indd 52 5/28/14 11:49 AM5/28/14 11:49 AM
Section 2.1 Transformations of Quadratic Functions 53
In Exercises 31–34, write a rule for g described by the
transformations of the graph of f. Then identify the
vertex.
(See Examples 3 and 4.)
31. f(x) = x
2
; vertical stretch by a factor of 4 and a
re ection in the x-axis, followed by a translation
2 units up
32. f(x) = x
2
; vertical shrink by a factor of
1
—
3
and a
re ection in the y-axis, followed by a translation
3 units right
33. f(x) = 8x
2
− 6; horizontal stretch by a factor of 2 and
a translation 2 units up, followed by a re ection in the
y-axis
34. f(x) = (x + 6)
2
+ 3; horizontal shrink by a factor of
1
—
2
and a translation 1 unit down, followed by a re ection
in the x-axis
USING TOOLS In Exercises 35–40, match the function
with its graph. Explain your reasoning.
35. g(x) = 2(x − 1)
2
− 2 36. g(x) =
1
—
2
( x + 1)
2
− 2
37. g(x) = −2(x − 1)
2
+ 2
38. g(x) = 2(x + 1)
2
+ 2 39. g(x) = −2(x + 1)
2
− 2
40. g(x) = 2(x − 1)
2
+ 2
A.
x
y
2
−4
−2
4
2−2−4
B.
x
y
4
2
−4
4
2−4
C.
x
y
−2
−4
4−2−4
D.
x
y
4
2
4−2−4
E.
x
y
4
−4
−2
42−2−4
F.
x
y
4
2
−4
42−2−4
JUSTIFYING STEPS In Exercises 41 and 42, justify each
step in writing a function g based on the transformations
of f(x) 2x
2
6x.
41. translation 6 units down followed by a re ection in
the x-axis
h(x) = f(x) − 6
= 2x
2
+ 6x − 6
g(x) = −h(x)
= −(2x
2
+ 6x − 6)
= −2x
2
− 6x + 6
42. re ection in the y-axis followed by a translation
4 units right
h(x) = f(−x)
= 2(−x)
2
+ 6(−x)
= 2(x)
2
− 6x
g(x) = h(x − 4)
= 2(x − 4)
2
+ 6(x − 4)
= 2x
2
− 10x + 8
43. MODELING WITH MATHEMATICS The function
h(x) = −0.03(x − 14)
2
+ 6 models the jump of a red
kangaroo, where x is the horizontal distance traveled
(in feet) and h(x) is the height (in feet). When the
kangaroo jumps from a higher location, it lands
5 feet farther away. Write a function that models the
second jump. (See Example 5.)
44. MODELING WITH MATHEMATICS The function
f(t) = −16t
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 at
the same time as on Earth?
54 Chapter 2 Quadratic Functions
Maintaining Mathematical Proficiency
Maintaining Mathematical Proficiency
A line of symmetry for the gure is shown in red. Find the coordinates of point A.
(Skills Review Handbook)
50.
x
y
(– 4, 3)
A
y = 1
51.
x
y
(0, 4) A
x = 2
52.
x
y
(2, – 2)
A
y = x
Reviewing what you learned in previous grades and lessons
45. MODELING WITH MATHEMATICS Flying sh use
their pectoral ns like airplane wings to glide through
the air.
a. Write an equation of the form y = a(x − h)
2
+ k
with vertex (33, 5) that models the ight path,
assuming the sh leaves the water at (0, 0).
b. What are the domain and range of the function?
What do they represent in this situation?
c. Does the value of a change when the ight path
has vertex (30, 4)? Justify your answer.
46. HOW DO YOU SEE IT? Describe the graph of g as a
transformation of the graph of f(x) = x
2
.
x
f
g
y
4
6
2
−2
2−4
−6
47. COMPARING METHODS Let the graph of g be a
translation 3 units up and 1 unit right followed by
a vertical stretch by a factor of 2 of the graph of
f(x) = x
2
.
a. Identify the values of a, h, and k and use vertex
form to write the transformed function.
b. Use function notation to write the transformed
function. Compare this function with your
function in part (a).
c. Suppose the vertical stretch was performed rst,
followed by the translations. Repeat parts (a)
and (b).
d. Which method do you prefer when writing a
transformed function? Explain.
48. THOUGHT PROVOKING A jump on a pogo stick
with a conventional spring can be modeled by
f(x) = −0.5(x − 6)
2
+ 18, where x is the horizontal
distance (in inches) and f(x) is the vertical distance
(in inches). Write at least one transformation of
the function and provide a possible reason for
your transformation.
49. MATHEMATICAL CONNECTIONS The area of a
circle depends on the radius, as shown in the graph.
Describe two different transformations of the graph
that model the area of the circle if the area is doubled.
Circle
Area (square feet)
Radius (feet)
r
A
20
10
0
30
210
43
A = r
2
π
