9.1
Graphing a Factorable
Quadratic Function
9.1
OBJECTIVES
1. Find the zeros for a factorable quadratic function
2. Find the vertex for the graph of a factorable
quadratic function
3. Sketch the graph of a factorable quadratic
function
If you know how fast a ball is thrown (its initial velocity) straight up into the air, you can
find its maximum height and predict the number of seconds it will be in the air. If you know
how long it was in the air, you can find both its maximum height and its initial velocity.
To analyze such a problem, we use a quadratic function.
Definitions: Quadratic Function
A function that can be written in the form
f(x) ax2 bx c, a 0
is called a quadratic function.
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The height of a ball (in feet) thrown up from the ground is determined by the quadratic
function
h(x) 16x2 vo x
in which vo represents the initial velocity and x represents the number of seconds that have
passed since the ball was thrown.
The function
h(x) 16x2 64x
gives us the height, after x seconds, of a ball thrown with an initial velocity of 64 ft/s.
671
672
CHAPTER 9
QUADRATIC EQUATIONS, FUNCTIONS, AND INEQUALITIES
Example 1
Finding the Zeros of a Quadratic Function
Find the zeros for the function
h(x) 16x2 64x
The zeros of the function are the values of x for which h(x) 0. We wish to solve the
equation
0 16x2 64x or
16x2 64x 0
In Section 6.9, we solved quadratic equations by the method of factoring. Using that technique, we find
16x(x 4) 0
The zero product rule tells us that there are two solutions, x 0 or x 4. Applying this
answer to our ball-throwing example tells us that the ball is at ground level twice, at 0 s
(just before the ball is thrown) and after 4 s (when the ball lands back on the ground).
CHECK YOURSELF 1
Find the zeros for the function
In our next example, we will sketch the graph of the function h(x).
Example 2
Graphing a Quadratic Function
NOTE This should not be
confused with a “sketch” of the
flight of the ball!
Sketch the graph of the function
h(x) 16x2 64x
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h(x) 16x2 48x
GRAPHING A FACTORABLE QUADRATIC FUNCTION
SECTION 9.1
673
From the table below, we can plot five points
x
h(x)
0
1
2
3
4
0
48
64
48
0
We will sketch our graph using only the first quadrant. We will plot the five points, then
connect them with a smooth curve.
80
70
(2, 64)
60
50
(1, 48)
(3, 48)
40
30
20
10
(0, 0)
0
(4, 0)
1
2
3
4
The shape of the graph of a quadratic function is called a parabola. Note that the curve at
the vertex is not angled or pointed.
CHECK YOURSELF 2
Sketch the graph of the function
h(x) 16x2 48x
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An important point for many mathematical applications is called the vertex.
Definitions: Vertex
The highest or lowest point on the graph for a quadratic function is called its
vertex.
In the case of the ball thrown into the air, the vertex is the highest point the ball reaches.
CHAPTER 9
QUADRATIC EQUATIONS, FUNCTIONS, AND INEQUALITIES
Example 3
Finding the Vertex
Find the vertex for the function
h(x) 16x2 64x
The vertex of the function is the ordered pair (x, h(x)) for which h(x) has the greatest value.
We will look at a table of values that will help us identify the vertex.
x
h(x)
0
1
2
3
4
0
48
64
48
0
Note that there seems to be a symmetric pattern. It takes the ball exactly as much time to
reach its vertex as it takes the ball to fall from its vertex to the ground. (This is because the
thing that is bringing the ball back to the ground, gravity, is constant).
The ball reaches its vertex exactly half way between its time of release and the time it
(0 4)
2 s. To find its height
falls to the ground. In this case, it reaches its vertex after
2
at the vertex, substitute 2 for the x.
h(2) 16(2)2 64(2)
64 128
64
The vertex is represented by the ordered pair (2, 64).
CHECK YOURSELF 3
Find the vertex for the function
h(x) 16x2 48x
The preceding example demonstrated a method for sketching the graph of any factorable quadratic function. This method is summarized in the following algorithm.
Step by Step: Sketching the Graph of a Quadratic
Function
Step 1 Factor the quadratic.
Step 2 Using the zero product rule, plot the points associated with the zeros of
the function on the x axis.
Step 3 Find the vertex (the mean of the two x values in step 2 and the function
value for that x) and plot the associated point.
Step 4 Draw a smooth curve connecting the three plotted points.
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674
GRAPHING A FACTORABLE QUADRATIC FUNCTION
SECTION 9.1
675
Example 4
Sketching the Graph of a Quadratic Function
Sketch the graph of the function
f(x) x2 4x 5
Factoring, we have
f(x) (x 1)(x 5)
Using the zero product rule, we find the two points (1, 0) and (5, 0). We plot those points.
y
(1, 0)
(5, 0)
x
To find the vertex, we find the mean of 1 and 5, which is 2, then we find f(2).
f(2) (2)2 4(2) 5
485
9
Plotting the vertex, (2, 9), and connecting the points with a smooth curve, we get
y
(5, 0)
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(1, 0)
(2, 9)
CHECK YOURSELF 4
Sketch the graph of the function
f(x) x2 2x 8
x
CHAPTER 9
QUADRATIC EQUATIONS, FUNCTIONS, AND INEQUALITIES
CHECK YOURSELF ANSWERS
1. (x 0) and (x 3)
2.
80
70
60
50
40
30
(2, 32)
(1, 32)
20
10
(0, 0)
0
3.
2, 36
3
4.
(3, 0)
1
3
2
4
y
x
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676
Name
9.1
Exercises
Section
Date
In exercises 1 to 12, find the zeros and the vertex of the given quadratic function.
1. f(x) x2 4x 3
2. f(x) x2 2x 24
3. f(x) x2 4x 12
4. f(x) 3x2 24x 36
1.
5. f(x) 4x2 16x 20
6. f(x) 2x2 10x 12
2.
7. f(x) 8x2 16x
8. f(x) 9x2 36x
3.
10. f(x) 6x2 24x
4.
12. f(x) x2 8x 12
5.
9. f(x) 5x2 20x
11. f(x) x2 2x 3
ANSWERS
In exercises 13 to 20, sketch the graph of the given quadratic function.
6.
13. f(x) x2 4x 3
7.
14. f(x) x2 x 2
y
y
8.
9.
x
x
10.
11.
12.
15. f(x) x2 2x 8
13.
16. f(x) 3x2 9x
y
y
14.
15.
16.
x
x
17.
18.
17. f(x) x2 x 6
18. f(x) x2 2x 3
y
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y
x
x
677
ANSWERS
19. f(x) x(x 6)
19.
20. f(x) 2x2 6x
y
y
20.
21. (a)
(b)
x
x
22. (a)
(b)
(c)
21. (a) Define a quadratic function that has zeros of 2 and 3. Express your function first
in factored form, and then in standard form ( f(x) ax 2 bx c).
(b) Define another quadratic function with the same zeros.
22. Let f(x) ax2 bx
(a) Write f(x) in factored form.
(b) Find the zeros of f(x).
(c) Find the x value of the vertex.
Answers
1. 3, 1; (2, 1)
3. 2, 6; (2, 16)
5. 5, 1; (2, 36)
7. 0, 2; (1, 8)
9. 0, 4; (2, 20)
11. 3, 1; (1, 4)
15.
y
y
x
17.
x
19.
y
x
y
x
21. (a) f(x) (x 2)(x 3), f(x) x 2 5x 6; (b) f(x) 2x 2 10x 12
678
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13.