10.2
Before
Now
Why?
Key Vocabulary
• minimum value
• maximum value
Graph
y 5 ax2 1 bx 1 c
You graphed simple quadratic functions.
You will graph general quadratic functions.
So you can investigate a cable’s height, as in Example 4.
You can use the properties below to graph any quadratic function. You will
justify the formula for the axis of symmetry in Exercise 38 on page 639.
For Your Notebook
KEY CONCEPT
Properties of the Graph of a Quadratic Function
The graph of y 5 ax2 1 bx 1 c is a parabola that:
• opens up if a . 0 and opens down if a , 0.
y 5 ax 2 1 bx 1 c, a > 0
• is narrower than the graph of y 5 x2
if ⏐a⏐ . 1 and wider if ⏐a⏐ , 1.
y
(0, c)
b
• has an axis of symmetry of x 5 2}
.
2a
x
b
• has a vertex with an x-coordinate of 2}
.
2a
b
x 5 2 2a
• has a y-intercept of c. So, the point (0, c) is
on the parabola.
EXAMPLE 1
Find the axis of symmetry and the vertex
Consider the function y 5 22x 2 1 12x 2 7.
a. Find the axis of symmetry of the graph of the function.
b. Find the vertex of the graph of the function.
Solution
a. For the function y 5 22x2 1 12x 2 7, a 5 22 and b 5 12.
b
IDENTIFY THE
VERTEX
Because the vertex
lies on the axis of
symmetry, x 5 3, the
x-coordinate of the
vertex is 3.
12
x 5 2}
5 2}
53
2a
2(22)
Substitute 22 for a and 12 for b. Then simplify.
b
2a
b. The x-coordinate of the vertex is 2} , or 3.
To find the y-coordinate, substitute 3 for x in the function and find y.
y 5 –2(3)2 1 12(3) 2 7 5 11
Substitute 3 for x. Then simplify.
c The vertex is (3, 11).
10.2 Graph y 5 ax2 1 bx 1 c
635
EXAMPLE 2
Graph y 5 ax 2 1 bx 1 c
Graph y 5 3x 2 2 6x 1 2.
STEP 1 Determine whether the parabola opens up or down. Because a . 0,
the parabola opens up.
AVOID ERRORS
26
b
Be sure to include the
negative sign before
the fraction when
calculating the axis of
symmetry.
STEP 2 Find and draw the axis of symmetry: x 5 2}
5 2}
5 1.
2a
2(3)
STEP 3 Find and plot the vertex.
y
b
(21, 11)
The x-coordinate of the vertex is 2}
,
2a
or 1.
To find the y-coordinate, substitute
1 for x in the function and simplify.
(3, 11)
x51
axis of
symmetry
y 5 3(1)2 2 6(1) 1 2 5 21
So, the vertex is (1, 21).
3
STEP 4 Plot two points. Choose two x-values
(0, 2)
less than the x-coordinate of the vertex.
Then find the corresponding y-values.
(2, 2)
3
REVIEW
REFLECTIONS
For help with
reflections, see p. 922.
x
0
21
y
2
11
STEP 5 Reflect the points plotted in Step 4 in the axis of symmetry.
STEP 6 Draw a parabola through the plotted points.
"MHFCSB
✓
vertex (1, 21)
at classzone.com
GUIDED PRACTICE
for Examples 1 and 2
1. Find the axis of symmetry and the vertex of the graph of the function
y 5 x2 2 2x 2 3.
2. Graph the function y 5 3x2 1 12x 2 1. Label the vertex and axis
of symmetry.
For Your Notebook
KEY CONCEPT
Minimum and Maximum Values
For y 5 ax2 1 bx 1 c, the y-coordinate of the vertex is the minimum value
of the function if a . 0 or the maximum value of the function if a , 0.
y 5 ax2 1 bx 1 c, a > 0
y 5 ax2 1 bx 1 c, a < 0
y
y
maximum
x
minimum
636
Chapter 10 Quadratic Equations and Functions
x
x
EXAMPLE 3
Find the minimum or maximum value
Tell whether the function f(x) 5 23x 2 2 12x 1 10 has a minimum value or a
maximum value. Then find the minimum or maximum value.
Solution
Because a 5 23 and 23 , 0, the parabola opens down and the function has a
maximum value. To find the maximum value, find the vertex.
b
212
b
x 5 2}
5 2}
5 22
2a
2(23)
The x-coordinate is 2}
2a .
f(22) 5 23(22)2 2 12(22) 1 10 5 22
Substitute 22 for x. Then simplify.
c The maximum value of the function is f(22) 5 22.
EXAMPLE 4
Find the minimum value of a function
SUSPENSION BRIDGES The suspension cables between the two towers of the
Mackinac Bridge in Michigan form a parabola that can be modeled by the
graph of y 5 0.000097x2 2 0.37x 1 549 where x and y are measured in feet.
What is the height of the cable above the water at its lowest point?
500
x
500
Solution
The lowest point of the cable is at the vertex of the parabola. Find the
x-coordinate of the vertex. Use a 5 0.000097 and b 5 20.37.
b
20.37
x 5 2}
5 2}
≈ 1910
2a
2(0.000097)
Use a calculator.
Substitute 1910 for x in the equation to find the y-coordinate of the vertex.
y ≈ 0.000097(1910)2 2 0.37(1910) 1 549 ≈ 196
c The cable is about 196 feet above the water at its lowest point.
✓
GUIDED PRACTICE
for Examples 3 and 4
3. Tell whether the function f(x) 5 6x2 1 18x 1 13 has a minimum value or a
maximum value. Then find the minimum or maximum value.
4. SUSPENSION BRIDGES The cables between the two towers of the Takoma
Narrows Bridge form a parabola that can be modeled by the graph of the
equation y 5 0.00014x2 2 0.4x 1 507 where x and y are measured in feet.
What is the height of the cable above the water at its lowest point? Round
your answer to the nearest foot.
10.2 Graph y 5 ax2 1 bx 1 c
637
10.2
EXERCISES
HOMEWORK
KEY
5 WORKED-OUT SOLUTIONS
on p. WS1 for Exs. 9 and 41
★ 5 STANDARDIZED TEST PRACTICE
Exs. 2, 12, 27, 37, 42, and 44
SKILL PRACTICE
1. VOCABULARY Explain how you can tell whether a quadratic function has
a maximum value or minimum value without graphing the function.
2.
★ WRITING Describe the steps you would take to graph a quadratic
function in standard form.
EXAMPLE 1
FINDING AXIS OF SYMMETRY AND VERTEX Find the axis of symmetry and
on p. 635
for Exs. 3–14
the vertex of the graph of the function.
3. y 5 2x2 2 8x 1 6
4. y 5 x2 2 6x 1 11
5. y 5 23x2 1 24x 2 22
6. y 5 2x2 2 10x
7. y 5 6x2 1 6x
8. y 5 4x2 1 7
2
3
12.
1
1 2
10. y 5 }
x 1 8x 2 9
9. y 5 2} x2 2 1
11. y 5 2}
x2 1 3x 2 2
4
2
★ MULTIPLE CHOICE What is the vertex of the graph of the function
y 5 23x2 1 18x 2 13?
A (23, 294)
B (23, 214)
C (3, 213)
D (3, 14)
ERROR ANALYSIS Describe and correct the error in finding the axis of
symmetry of the graph of the given function.
3
2
13. y 5 2x2 1 16x 2 1
14. y 5 2}x2 1 18x 2 5
18
16
b
5}
54
x5}
b
x 5 2}
5 2}
5 –6
3
The axis of symmetry is x 5 4.
The axis of symmetry is x 5 26.
2a
21 } 2
2a
2(2)
2
EXAMPLE 2
GRAPHING QUADRATIC FUNCTIONS Graph the function. Label the vertex
on p. 636
for Exs. 15–27
and axis of symmetry.
15. y 5 x2 1 6x 1 2
16. y 5 x2 1 4x 1 8
17. y 5 2x2 1 7x 1 21
18. y 5 5x2 1 10x 2 3
19. y 5 4x2 1 x 2 32
20. y 5 24x2 1 4x 1 8
21. y 5 23x2 2 2x 2 5
22. y 5 28x2 2 12x 1 1
1
1
23. y 5 2x2 1 }
x1}
1 2
24. y 5 }
x 1 6x 2 9
3
1
25. y 5 2} x2 1 6x 1 3
2
27.
4
2
1 2
26. y 5 2}
x 2x11
4
★ MULTIPLE CHOICE Which function has the
y
graph shown?
A y 5 22x2 1 8x 1 3
(2, 5)
5
(0, 3)
1
B y 5 2}
x2 1 2x 1 3
2
1 2
C y5}
x 1 2x 1 3
2
2
D y 5 2x 1 8x 1 3
638
Chapter 10 Quadratic Equations and Functions
1
x
EXAMPLE 3
on p. 637
for Exs. 28–36
MAXIMUM AND MINIMUM VALUES Tell whether the function has a minimum
value or a maximum value. Then find the minimum or maximum value.
28. f(x) 5 x2 2 6
29. f(x) 5 25x2 1 7
30. f(x) 5 4x2 1 32x
31. f(x) 5 23x2 1 12x 2 20
32. f(x) 5 x2 1 7x 1 8
33. f(x) 5 22x2 2 x 1 10
1 2
34. f(x) 5 }
x 2 2x 1 5
35. f(x) 5 2} x2 1 9x
2
37.
3
8
1 2
36. f(x) 5 }
x 1 7x 1 11
4
★ WRITING Compare the graph of y 5 x 1 4x 1 1 with the graph of
y 5 x2 2 4x 1 1.
2
38. REASONING Follow the steps below to justify the equation for the axis
of symmetry for the graph of y 5 ax2 1 bx 1 c. Because the graph of
y 5 ax2 1 bx 1 c is a vertical translation of the graph of y 5 ax2 1 bx,
the two graphs have the same axis of symmetry. Use the function
y 5 ax2 1 bx in place of y 5 ax2 1 bx 1 c.
a. Find the x-intercepts of the graph of y 5 ax2 1 bx. (You can do this
by finding the zeros of the function y 5 ax2 1 bx using factoring.)
b. Because a parabola is symmetric about its axis of symmetry, the
axis of symmetry passes through a point halfway between the
x-intercepts of the parabola. Find the x-coordinate of this point.
What is an equation of the vertical line through this point?
39. CHALLENGE Write a function of the form y 5 ax2 1 bx whose graph
contains the points (1, 6) and (3, 6).
PROBLEM SOLVING
GRAPHING CALCULATOR You may wish to use a graphing calculator to
complete the following Problem Solving exercises.
EXAMPLE 4
on p. 637
for Exs. 40–42
40. SPIDERS Fishing spiders can propel themselves across water and leap
vertically from the surface of the water. During a vertical jump, the
height of the body of the spider can be modeled by the function
y 5 24500x2 1 820x 1 43 where x is the duration (in seconds) of the
jump and y is the height (in millimeters) of the spider above the surface
of the water. After how many seconds does the spider’s body reach its
maximum height? What is the maximum height?
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
41. ARCHITECTURE The parabolic arches that support the roof of the
Dallas Convention Center can be modeled by the graph of the equation
y 5 20.0019x2 1 0.71x where x and y are measured in feet. What is the
height h at the highest point of the arch as shown in the diagram?
y
h
20
30
x
GPSQSPCMFNTPMWJOHIFMQBUDMBTT[POFDPN
10.2 Graph y 5 ax2 1 bx 1 c
639
42.
★ EXTENDED RESPONSE Students are selling packages of flower bulbs
to raise money for a class trip. Last year, when the students charged
$5 per package, they sold 150 packages. The students want to increase
the cost per package. They estimate that they will lose 10 sales for each
$1 increase in the cost per package. The sales revenue R (in dollars)
generated by selling the packages is given by the function
R 5 (5 1 n)(150 2 10n) where n is the number of $1 increases.
a. Write the function in standard form.
b. Find the maximum value of the function.
c. At what price should the packages be sold to generate the most sales
revenue? Explain your reasoning.
y
43. AIRCRAFT An aircraft hangar is a large
building where planes are stored. The
opening of one airport hangar is a
parabolic arch that can be modeled by the
graph of the equation y 5 20.007x2 1 1.7x
where x and y are measured in feet. Graph
the function. Use the graph to determine
how wide the hangar is at its base.
44.
50
x
50
★ SHORT RESPONSE The casts of some Broadway shows go on tour,
performing their shows in cities across the United States. For the period
1990–2001, the number of tickets sold S (in millions) for Broadway road
tours can be modeled by the function S 5 332 1 132t 2 10.4t 2 where t is
the number of years since 1990. Was the greatest number of tickets for
Broadway road tours sold in 1995? Explain.
45. CHALLENGE During an archery competition,
an archer shoots an arrow from 1.5 meters
off of the ground. The arrow follows the
parabolic path shown and hits the ground in
front of the target 90 meters away. Use the
y-intercept and the points on the graph to
write an equation for the graph that models
the path of the arrow.
y
2
vertex (18, 1.6)
(0, 1.5)
1
10
(90, 0) x
MIXED REVIEW
Graph the equation. (pp. 215, 225, 244)
47. x 2 5y 5 15
2
48. y 5 2}
x26
49. 23(4 2 2x) 2 9 (p. 96)
50. 2(1 2 a) 2 5a (p. 96)
51. } (p. 103)
52. (22mn)4 (p. 489)
53. 5 p (7w 7)2 (p. 489)
46. y 5 3
3
Simplify.
12y 2 4
24
6u3 uv2
54. } p } (p. 495)
v
36
PREVIEW
Find the zeros of the polynomial function.
Prepare for
Lesson 10.3 in
Exs. 55–58.
55. f(x) 5 x2 2 4x 2 21 (p. 583)
56. f(x) 5 x2 1 10x 1 24 (p. 583)
57. f(x) 5 5x2 1 18x 1 9 (p. 593)
58. f(x) 5 2x2 1 4x 2 6 (p. 593)
640
EXTRA PRACTICE for Lesson 10.2, p. 947
ONLINE QUIZ at classzone.com
Extension
Use after Lesson 10.2
Graph Quadratic Functions in
Intercept Form
GOAL Graph quadratic functions in intercept form.
Key Vocabulary
• intercept form
In Lesson 10.2 you graphed quadratic functions written in standard
form. Quadratic functions can also be written in intercept form,
y 5 a(x 2 p)(x 2 q) where a Þ 0. In this form, the x-intercepts of the graph
can easily be determined.
For Your Notebook
KEY CONCEPT
Graph of Intercept Form y 5 a(x 2 p)(x 2 q)
Characteristics of the graph of y 5 a(x 2 p)(x 2 q):
y
• The x-intercepts are p and q.
x5
p1q
2
• The axis of symmetry is halfway between
(p, 0) and (q, 0). So, the axis of symmetry
p1q
2
is x 5 }.
(q, 0)
(p, 0)
x
• The parabola opens up if a . 0 and opens
down if a , 0.
EXAMPLE 1
Graph a quadratic function in intercept form
Graph y 5 2(x 1 1)(x 2 5).
FIND ZEROS OF A
FUNCTION
Notice that the x-intercepts
of the graph are also the
zeros of the function:
0 5 2(x 1 1)( x 2 5)
x 1 1 5 0 or x 2 5 5 0
x 5 21 or x 5 5
Solution
STEP 1 Identify and plot the x-intercepts. Because p 5 21 and q 5 5,
the x-intercepts occur at the points (21, 0) and (5, 0).
STEP 2 Find and draw the axis of symmetry.
y
p1q
15
x 5 } 5 21
}52
2
2
(2, 9)
STEP 3 Find and plot the vertex.
The x-coordinate of the vertex is 2.
5
To find the y-coordinate of the vertex,
substitute 2 for x and simplify.
y 5 2(2 1 1)(2 2 5) 5 9
So, the vertex is (2, 9).
STEP 4 Draw a parabola through the vertex
(21, 0)
(5, 0)
1
x
and the points where the x-intercepts
occur.
Extension: Graph Quadratic Functions in Intercept Form
641
EXAMPLE 2
Graph a quadratic function
Graph y 5 2x 2 2 8.
Solution
STEP 1 Rewrite the quadratic function in intercept form.
y 5 2x2 2 8
2
Write original function.
5 2(x 2 4)
Factor out common factor.
5 2(x 1 2)(x 2 2)
Difference of two squares pattern
STEP 2 Identify and plot the x-intercepts. Because p 5 22 and q 5 2,
the x-intercepts occur at the points (22, 0) and (2, 0).
STEP 3 Find and draw the axis of symmetry.
p1q
12
x 5 } 5 22
}50
2
2
y
(22, 0)
21
1
(2, 0)
STEP 4 Find and plot the vertex.
The x-coordinate of the vertex is 0.
The y-coordinate of the vertex is:
y 5 2(0)2 2 8 5 28
So, the vertex is (0, 28).
(0, 28)
STEP 5 Draw a parabola through the vertex and
the points where the x-intercepts occur.
"MHFCSB
at classzone.com
PRACTICE
EXAMPLE 1
on p. 641 for
Exs. 1–9
Graph the quadratic function. Label the vertex, axis of symmetry, and
x-intercepts.
1. y 5 (x 1 2)(x 2 3)
2. y 5 (x 1 5)(x 1 2)
3. y 5 (x 1 9)2
4. y 5 22(x 2 5)(x 1 1)
5. y 5 25(x 1 7)(x 1 2)
6. y 5 3(x 2 6)(x 2 3)
1
7. y 5 2}
(x 1 4)(x 2 2)
8. y 5 (x 2 7)(2x 2 3)
9. y 5 2(x 1 10)(x 2 3)
2
EXAMPLE 2
10. y 5 2x2 1 8x 2 16
11. y 5 2x2 2 9x 2 18
12. y 5 12x2 2 48
on p. 642 for
Exs. 10–15
13. y 5 26x2 1 294
14. y 5 3x2 2 24x 1 36
15. y 5 20x2 2 6x 2 2
16. Follow the steps below to write an equation of the
y
parabola shown.
a. Find the x-intercepts.
b. Use the values of p and q and the coordinates of
the vertex to find the value of a in the equation
y 5 a(x 2 p)(x 2 q).
c. Write a quadratic equation in intercept form.
642
Chapter 10 Quadratic Equations and Functions
1
1
x
x