8-3
Graphing Quadratic Functions
Going Deeper
Essential question: How can you describe key attributes of the graph of
f(x) = ax2 + bx + c by analyzing its equation?
Standards for
Mathematical Content
• If you know the coordinates of a point on a
parabola, how can you find the coordinates of
its reflection across the axis of symmetry? The
F-IF.3.7 Graph functions expressed symbolically
and show key features of the graph ...*
F-IF.3.7a Graph … quadratic functions and show
intercepts, maxima, and minima.*
F-IF.3.8a Use the process of factoring … in a
quadratic function to show zeros …
Prerequisites
reflection will have the same y-value as the given
point and be the same distance but opposite
direction from the axis of symmetry.
EXTRA EXAMPLE
Graph the function f(x) = x2 - 4x - 5 by factoring.
f(x) = (x + 1) (x - 5)
4
2
Transformations of the graph of f(x) = x
y
2
Factoring ax2 + bx + c
x
-4
Math Background
-2
Students will graph quadratic functions in standard
form. They will describe key attributes of the graph
(vertex, maximum or minimum, and intercepts)
by analyzing the equation. They will use these
attributes and factoring to draw the graphs.
0
2
4
6
8
-4
-6
-8
CLOS E
Review factoring of trinomials with students,
including the special cases. Then ask students to set
these factored trinomials equal to 0 and ask what
values of x make these equations true. Ask students
what these x-values represent in terms of the graph
of the function.
Essential Question
How can you describe key attributes of the graph of
f(x) = ax2 + bx + c by analyzing its equation?
Factor the quadratic trinomial and solve f (x) = 0
to find the x-intercepts. Find the x-value halfway
between the zeros of the function to determine the
axis of symmetry and use that x-value to find the
vertex. To find the y-intercept, find f (0).
T EACH
1
Summarize
Have student write a journal entry in which they
describe how to graph a quadratic function in
standard form by using factoring to find key
attributes of the graph.
EXAMPLE
Questioning Strategies
• How does factoring generate the x-intercepts
of the function’s graph? The points where the
graph of f(x) crosses the x-axis have y-values that
are 0, so the x-intercepts are found by solving the
equation f(x) = 0, which can be done by factoring
f(x) and applying the zero-product property.
PR ACTICE
Where skills are
taught
• If you fold the parabola along its axis of
symmetry, where would the two sides of the
parabola be in relation to each other? One side of
1 EXAMPLE
Where skills are
practiced
EX. 1–6
the parabola would coincide with the other side.
Chapter 8
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Lesson 3
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IN T RO DUC E
Name
Class
Notes
8-3
Date
Graphing Quadratic Functions
Going Deeper
Essential question: How can you describe key attributes of the graph of
f (x) = ax2 + bx + c by analyzing its equation?
To graph a function of the form f (x) = ax2 + bx + c, called standard form, you can
analyze the key features of the graph by factoring.
1
F-IF.3.7a
EXAMPLE
Graphing f (x) = x2 + bx + c
Graph the function f (x) = x2 + 2x - 3 by factoring.
You can determine the x-intercepts of the graph by factoring to solve f (x) = 0:
A
f (x) = x2 + 2x - 3 = ( x - 1
-3
x=
)( x +
1
3 ), so f (x) = 0 when x =
or
.
© Houghton Mifflin Harcourt Publishing Company
The graph of f (x) = x2 + 2x - 3 intersects the x-axis at ( 1 , 0 ) and ( -3 , 0 ).
B
The axis of symmetry of the graph is a vertical line that is halfway between the two
x-intercepts and passes through the vertex. The axis of symmetry is x = -1 . So, the
vertex is ( -1 , -4 ).
C
Find another point on the graph and reflect it across the axis of symmetry. Use the
point (2, 5). The x-value is 3 units from the axis of symmetry, so its reflection is
( -4 , 5 ).
D
Use the five points to graph the function.
y
4
2
x
-4
0
-2
2
4
-2
REFLECT
1a. A useful point to plot is where the y-intercept occurs. How can you find
the y-intercept? What point is the reflection across the axis of symmetry of
the point where the y-intercept occurs?
The y-intercept is f (0) = c. In this case, f (0) = -3.
The reflection of (0, -3) across the axis of symmetry is (-2, -3).
427
Chapter 8
Lesson 3
PRACTICE
Write the rule for the quadratic function in the form you would use to graph it.
Then graph the function.
f (x) = x2 + 4x + 3
2.
f (x) = x2 - 2x - 3
f(x) = (x - 3) (x + 1)
f(x) = (x + 3) (x + 1)
y
4
4
y
2
x
0
-4
2
x
4
-4
2
4
-2
-4
3.
0
-2
-2
-4
4.
2
f (x) = x - 4
f (x) = x - 6x + 5
f(x) = (x - 1) (x - 5)
f(x) = (x - 2) (x + 2)
4
2
y
4
2
y
2
x
-4
-2
0
2
x
0
4
4
8
-2
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
1.
-4
5.
2
6.
f (x) = x + 2x - 8
8
f (x) = x2 - 7x + 10
f(x) = (x - 5) (x - 2)
f(x) = (x + 4) (x - 2)
y
8
4
y
4
x
-8
Chapter 8
Chapter 8
-4
0
4
x
0
8
-4
-4
-8
-8
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4
8
Lesson 3
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Lesson 3
3. 18 feet; 1.5 seconds; 3 seconds
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. x = -2; (-2, -8); -4; Possible answers:
(1, 1) and (2, 8)
Problem Solving
1.
2. x = 1; (1, 8); 6; Possible answers:
(−1, 0) and (-2, -10)
4 seconds
© Houghton Mifflin Harcourt Publishing Company
2.
36 ft; 1.5 seconds
3. H
4. C
5. C
Chapter 8
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Lesson 3
Name
Class
Date
Notes
8-3
Additional Practice
© Houghton Mifflin Harcourt Publishing Company
429
Chapter 8
Lesson 3
© Houghton Mifflin Harcourt Publishing Company
© Houghton Mifflin Harcourt Publishing Company
Problem Solving
Chapter 8
Chapter 8
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Lesson 3
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Lesson 3