1. No category

8-2
Characteristics of Quadratic
Functions
Connection: Connecting f (x) = x 2 to
g (x) = (x - h) 2 + k
Essential question: What are the effects of the constants h and k on the graph
of g(x) = (x - h)2 + k?
IN TR OD UCE
Standards for
Mathematical Content
Show students the graph of f (x) = x. Then show
them graphs of g (x) = x + b with b both positive
and negative. Ask them to describe the effect b has
on the graph of f (x) = x. Do a similar activity with
the graphs of f (x) = |x|, g (x) = |x| + k, m(x) = |x - h|
and n(x) = |x - h| + k. Ask students how they think
h and k will affect the graph of g (x) = (x - h)2 + k.
A-CED.1.2 ... graph equations on coordinate axes
with labels and scales.*
F-IF.1.2 Use function notation, evaluate functions
for inputs in their domains ...
F-IF.2.4 For a function . . . interpret key features of
graphs and tables in terms of the quantities ...*
F-IF.2.5 Relate the domain of a function to its
graph and, where applicable, to the quantitative
relationship it describes.*
F-IF.3.7a Graph . . . quadratic functions and show
intercepts, maxima, and minima.*
F-BF.1.1 Write a function that describes a
relationship between two quantities.*
F-BF.2.3 Identify the effect on the graph of
replacing f (x) by f (x) + k ... and f (x + k) for specific
values of k (both positive and negative); find the
value of k given the graphs.
TE ACH
1
Questioning Strategies
• How is the graph of g (x) = x2 + 2 related to the
graph of g (x) = x2 - 2? They are both translated
graphs of the same parent graph f (x) = x2, but
g (x) = x2 + 2 is translated 2 units up and g (x) =
x2 - 2 is translated 2 units down.
Functions
Quadratic Functions
EXTRA EXAMPLE
Graph each quadratic function. How is each related
to the graph of f (x) = x2?
Math Background
In this lesson, students will graph quadratic
functions of the form f (x) = x2 + k. This graph also
will be symmetric in the y-axis, but its vertex will be
(0, k). Students also will graph quadratic functions
of the form f (x) = (x - h)2. This graph will be
translated h units horizontally with vertex (h, 0). It
will not be symmetric in the y-axis. Finally, students
will write equations for functions of the form
f(x) = (x - h)2 + k and make the connection
between the equation of the function and the vertex
(h, k) of its graph.
A. g (x) = x2 + 3
The graph is translated 3 units up from the
graph of the parent function f (x) = x2. The
graph opens upward with the vertex at (0, 3).
B. g (x) = x2 - 3
The graph is translated 3 units down from the
graph of the parent function f (x) = x2. The
graph opens upward with the vertex at (0, -3).
419
Lesson 2
© Houghton Mifflin Harcourt Publishing Company
• Is the vertex of the graph of g(x) = x2 + 2 the same
as the vertex of the graph of g(x) = x2 - 2? No;
g (x) = x2 + 2 has vertex (0, 2), and g (x) = x2 - 2
has vertex (0, -2).
Prerequisites
Chapter 8
EXAMPLE
Name
Class
Notes
8-2
Date
Characteristics of Quadratic Functions
Connection: Connecting f (x ) = x 2 to g (x ) = (x - h )2 + k
Essential question: What is the effect of the constants h and k on the
graph of g(x) = (x - h)2 + k?
F-BF.2.3
1
EXAMPLE
Graphing Functions of the Form g(x) = x2 + k
Graph each quadratic function. (The graph of the parent function f (x) = x2
is shown in gray.)
A
g(x) = x2 + 2
x
g(x) = x 2 + 2
-3
11
-2
6
10
y
8
3
0
2
1
3
2
6
3
11
6
4
2
x
-4
0
-2
2
4
2
4
2
g(x) = x - 2
© Houghton Mifflin Harcourt Publishing Company
B
-1
x
g(x) = x2 - 2
-3
7
6
-2
2
4
-1
-1
0
-2
1
-1
2
2
3
7
y
2
x
-4
0
-2
-2
-4
419
Chapter 8
Lesson 2
REFLECT
1a. How is the graph of g(x) = x2 + 2 related to the graph of f (x) = x2?
Vertical translation of 2 units up
1b. How is the graph of g(x) = x2 - 2 related to the graph of f (x) = x2?
Vertical translation of k units (up if k > 0, down if k < 0)
2
F-BF.2.3
EXAMPLE
Graphing Functions of the Form g(x) = (x - h)2
Graph each quadratic function. (The graph of the parent function f (x) = x2
is shown in gray.)
A
B
g(x) = (x - 1)2
x
g(x) = (x - 1) 2
-2
9
-1
4
0
1
4
1
0
2
2
1
3
4
4
9
8
y
6
x
-4
-2
0
2
4
2
4
-2
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Vertical translation of 2 units down
1c. In general, how is the graph of g(x) = x2 + k related to the graph of f (x) = x2?
g(x) = (x + 1)2
x
g(x) = (x + 1) 2
-4
9
-3
4
8
y
6
Chapter 8
Chapter 8
-2
1
-1
0
0
1
1
4
2
9
4
2
x
-4
-2
0
-2
420
Lesson 2
420
Lesson 2
2
3
EXAMPLE
EXAMPLE
Questioning Strategies
• How is the vertex of the graph of g(x) = (x - h)2 + k
related to the vertex of the graph of f (x) = x2?
The graph of f(x) = x2 has vertex (0, 0), while
the graph of g(x) = (x - h)2 + k has vertex (h, k).
Questioning Strategies
• How is the graph of g (x) = (x - 1)2 related to the
graph of g (x) = (x + 1)2? They are both translated
graphs of the same parent graph f (x) = x2, but
the graph of g (x) = (x - 1)2 is translated 1 unit
• How is the graph of g (x) = (x - h)2 + k related to
the graph of f (x) = x2? The graph of g (x) =
(x - h)2 + k is the same shape as the graph of
f (x) = x2, but it is translated k units vertically
to the right and has vertex (1, 0), while the graph
of g (x) = (x + 1)2 is translated 1 unit to the left
and has vertex (-1, 0).
• What is the vertex of g (x) = (x - h)2? (h, 0)
and h units horizontally.
EXTRA EXAMPLE
Graph each quadratic function. How is each related
to the graph of f (x) = x2?
EXTRA EXAMPLE
Write the equation for the quadratic function whose
graph is shown.
A. g (x) = (x - 4)2
The graph is translated 4 units right from the
graph of the parent function f (x) = x2. The
graph opens upward with the vertex at (4, 0).
10
y
8
B. g (x) = (x + 4)2
6
The graph is translated 4 units left from the
graph of the parent function f (x) = x2. The
graph opens upward with the vertex at (-4, 0).
4
2
x
-4
Highlighting the
Standards
0
2
4
-2
A. Compare the graph to the graph of f (x) = x2.
1 EXAMPLE
and 2 EXAMPLE include
opportunities to address Mathematical
Practice Standard 7 (Look for and make use of
structure). Students should realize that adding
k to x2 moves the graph up for k > 0 or down
for k < 0 and that subtracting h from x moves
the graph left for h < 0 or right for h > 0.
This is true for all nonzero values of k and h.
Recognizing this structure allows students to
write equations from graphs in 3 EXAMPLE .
The graph is translated 3 units up and 1 unit
left; the vertex is at (-1, 3) instead of (0, 0).
B. Determine the values of h and k for the function
g (x) = (x - h)2 + k and write the function for the
graph. h = -1; k = 3; g (x) = (x + 1)2 + 3
421
Lesson 2
© Houghton Mifflin Harcourt Publishing Company
Chapter 8
-2
Notes
REFLECT
2a. How is the graph of g(x) = (x - 1)2 related to the graph of f (x) = x2?
Horizontal translation of 1 unit to the right
2b. How is the graph of g(x) = (x + 1)2 related to the graph of f (x) = x2?
Horizontal translation of 1 unit to the left
2c. In general, how is the graph of g(x) = (x - h)2 related to the graph of f (x) = x2?
Horizontal translation of h units (to the right if h > 0, to the left if h < 0)
F-BF.1.1
3
EXAMPLE
Writing Equations for Quadratic Functions
Write the equation for the quadratic function whose graph is shown.
A
y
8
Compare the given graph to the graph
of the parent function f (x) = x2.
6
Complete the table below to describe
how the parent function must be translated to
get the graph shown here.
4
2
x
0
-2
2
4
6
8
© Houghton Mifflin Harcourt Publishing Company
-2
Type of Translation
Number of Units
Direction
Horizontal Translation
3
To the right
Vertical Translation
2
Up
Determine the values of h and k for the function g(x) = (x - h)2 + k.
B
t h is the number of units that the parent function is translated horizontally. For a
translation to the right, h is positive; for a translation to the left, h is negative.
t k is the number of units that the parent function is translated vertically. For a
translation up, k is positive; for a translation down, k is negative.
3
So, h =
and k =
2
. The equation is
g(x) = (x - 3)2 + 2
.
REFLECT
3a. What can you do to check that your equation is correct?
Graph the equation to see if the graph is identical to the given one.
421
Chapter 8
Lesson 2
3b. If the graph of a quadratic function is a translation of the graph of the parent
function, explain how you can use the vertex of the translated graph to help you
determine the equation for the function.
If the vertex has coordinates (h, k), then the equation for the function
3c. Error Analysis A student says that the graph of g(x) = (x + 2)2 + 1 is the graph
of the parent function translated 2 units to the right and 1 unit up. Explain what is
incorrect about this statement.
When written in the form g(x) = (x - h)2 + k, the function is g(x) = (x - (-2))2 + 1,
so the parent graph is translated 2 units to the left and 1 unit up.
PRACTICE
Graph each quadratic function.
1.
f (x) = x2 + 4
10
2.
f (x) = x2 - 5
y
y
2
x
8
-5
-3
6
-1 0
1
3
5
-2
4
-6
2
x
-4
3.
-2
0
2
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
is g(x) = (x - h)2 + k.
-8
4
f (x) = (x - 2)2
4.
f (x) = (x + 3)2
y
10
6
y
8
4
2
4
x
-2
0
2
4
6
2
8
x
-2
Chapter 8
Chapter 8
-8
422
-6
-4
-2
0
2
Lesson 2
422
Lesson 2
CLOSE
PR ACTICE
Essential Question
What are the effects of the constants h and k on the
graph of g(x) = (x - h)2 + k?
Where skills are
taught
The constant h moves the graph of the parent
function f(x) = x2 right h units if h > 0 or h units
left if h < 0. The constant k moves the graph up
k units if k > 0 or down k units if k < 0. The vertex
of the graph is (h, k).
Where skills are
practiced
1 EXAMPLE
EXS. 1, 2
2 EXAMPLE
EXS. 3, 4
3 EXAMPLE
EXS. 9–12
Exercises 5-8: Students extend what they learned
in 3 EXAMPLE to graph quadratic functions of the
form g (x) = (x - h)2 + k.
Summarize
Have students write a journal entry in which they
describe the effects of the constants h and k on the
graph of g (x) = (x - h)2 + k by comparing its graph
to the graph of the parent function f (x) = x2.
Exercises 13-18: Students use their knowledge of
transformations to find the domain, range, vertex,
and axis of symmetry of quadratic functions of the
form g (x) = (x - h)2 + k. Students should see that
the domains do not change, but the ranges do.
Exercise 19: Students should see that a quadratic
function of the form f (x) = x2 - 1 is an even
function, while f (x) = (x - 1)2 is not. Students
should realize that a vertical translation of f (x) = x2
results in an even function, but that a horizontal
translation does not.
© Houghton Mifflin Harcourt Publishing Company
Chapter 8
423
Lesson 2
Notes
5. f (x) = (x - 5)2 - 2
8
6. f (x) = (x - 1)2 + 1
y
10
6
8
4
6
y
2
x
0
-2
2
5
2
8
x
-2
-4
7. f (x) = (x + 4)2 + 3
0
-2
2
4
6
8. f (x) = (x + 2)2 - 4
y
8
5
6
y
3
4
1
-7
2
x
0
-1
-1
-5
1
3
4
6
x
-8
-6
-4
-2
0
2
-5
-2
Write a rule for the quadratic function whose graph is shown.
9.
10.
y
8
y
6
2
4
© Houghton Mifflin Harcourt Publishing Company
x
-2 0
-2
2
4
6
2
8
x
-4
-2
-4
0
2
-2
f(x) = (x - 2)2 - 3
11.
8
f(x) = (x - 1)2 + 4
12.
y
8
6
y
6
4
2
2
x
-8
-6
-4
0
x
2
-8
-6
-4
0
-2
-2
2
-2
f(x) = (x + 3)2 - 1
f(x) = (x + 5)2 + 4
423
Chapter 8
Lesson 2
Determine the domain, range, vertex, and axis of symmetry of the function.
14. f (x) = x2 + 4
15. f (x) = (x + 5)2
D = {real nos.};
D = {real nos.};
D = {real nos.};
R = {y | y ≥ 0}
R = {y | y ≥ 4}
R = {y | y ≥ 0}
vertex: (3, 0)
vertex: (0, 4)
vertex: (-5, 0)
axis of sym: x = 3
axis of sym: x = 0
axis of sym: x = -5
2
17. f (x) = (x + 1) - 6
18. f (x) = (x - 2)2 + 8
D = {real nos.};
D = {real nos.};
D = {real nos.};
R = {y | y ≥ -7}
R = {y | y ≥ -6}
R = {y | y ≥ 8}
vertex: (0, -7)
vertex: (-1, -6)
vertex: (2, 8)
axis of sym: x = 0
axis of sym: x = -1
axis of sym: x = 2
16. f (x) = x - 7
2
19. A function is called even if f (-x) = f (x) for all x in the domain of the function. For
instance, if f (x) = x2, then f (-x) = (-x)2 = x2 = f (x). In other words, you get the
same value when you square -x as you do when you square x. So, f (x) = x2 is an
even function.
a. Is f (x) = x2 - 1 an even function? Explain.
Yes; f (-x) = (-x)2 - 1 = x2 - 1 = f (x)
b. Is f (x) = (x - 1)2 an even function? Explain.
No; f (-x) = (-x - 1)2 = (-(x + 1))2 = (x + 1)2 ≠ f (x)
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
13. f (x) = (x - 3)2
Chapter 8
Chapter 8
424
Lesson 2
424
Lesson 2
ADD I T I O N A L P R AC T I C E
AND PRO BL E M S O LV I N G
Assign these pages to help your students practice
and apply important lesson concepts. For
additional exercises, see the Student Edition.
Answers
Additional Practice
1. y = (x + 3.5)2 – 6.25
2. y = (x – 2)2 + 5
3. y = (x – 5)2
4. y = –(x – 3.5)2 + 2.25
5. y = (x – 3)2 – 4
6. y = –(x + 1)2 + 1
7. D: all real numbers; R: y ≥ 1; (1, 1); x = 1
8. D: all real numbers; R: y ≤ 4; (2, 4); x = 2
9. D: all real numbers; R: y ≥ –3; (–5, –3); x = –5
10. D: all real numbers; R: y ≥ –5; (1, –5); x = 1
© Houghton Mifflin Harcourt Publishing Company
11. D: all real numbers; R: y ≤ –22; (–2, –22);
x = –2
12. D: all real numbers; R: y ≥ –36; (–1, –36);
x = –1
Problem Solving
1. Yes; the vertex is (6, 620) and 620 > 612
2. f(x) = –(x – 25)2 + 625; x = 25
3. (10, 8)
4. B
5.
H
6. D
7.
G
Chapter 8
425
Lesson 2
Name
Class
Notes
8-2
Date
© Houghton Mifflin Harcourt Publishing Company
Additional Practice
425
Chapter 8
Lesson 2
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Chapter 8
Chapter 8
426
Lesson 2
426
Lesson 2