1. Real World Problem Solving Quadratic Equations 8
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. Example 5: Application
The graph of f(x) = –0.06x x can be used to model the height in meters of an arch support for a bridge, where the x-axis represents the water level and x represents the horizontal distance in meters from where the arch support enters the water. Can a sailboat that is 14 meters tall pass under the bridge? Explain.
The vertex represents the highest point of the arch support.

3. Example 5 Continued
Step 1 Find the x-coordinate.
a = – 0.06, b = 0.6
Identify a and b.
Substitute –0.06 for a and 0.6 for b.
Step 2 Find the corresponding y-coordinate.
Use the function rule.
f(x) = –0.06x x
= –0.06(5) (5)
Substitute 5 for x.
= 11.76
Since the height of each support is m, the sailboat cannot pass under the bridge.

4. Check It Out! Example 5
The height of a small rise in a roller coaster track is modeled by f(x) = –0.07x x , where x is the distance in feet from a supported pole at ground level. Find the greatest height of the rise.
Step 1 Find the x-coordinate.
a = – 0.07, b= 0.42
Identify a and b.
Substitute –0.07 for a and 0.42 for b.

5. Check It Out! Example 5 Continued
Step 2 Find the corresponding y-coordinate.
f(x) = –0.07x x
Use the function rule.
= –0.07(3) (3)
Substitute 3 for x.
= 7 ft
The height of the rise is 7 ft.

6. Example 2: Application
The height in feet of a basketball that is thrown can be modeled by f(x) = –16x2 + 32x, where x is the time in seconds after it is thrown. Find the basketball’s maximum height and the time it takes the basketball to reach this height. Then find how long the basketball is in the air.

7. List the important information:
Example 2 Continued 1
Understand the Problem
The answer includes three parts: the maximum height, the time to reach the maximum height, and the time to reach the ground.
List the important information:
The function f(x) = –16x2 + 32x models the height of the basketball after x seconds.

8. Example 2 Continued 2
Make a Plan
Find the vertex of the graph because the maximum height of the basketball and the time it takes to reach it are the coordinates of the vertex. The basketball will hit the ground when its height is 0, so find the zeros of the function. You can do this by graphing.

9. Example 2 Continued
Solve 3
Step 1 Find the axis of symmetry.
Use x = Substitute
–16 for a and 32 for b.
Simplify.
The axis of symmetry is x = 1.

10. Example 2 Continued
Step 2 Find the vertex.
f(x) = –16x2 + 32x
The x-coordinate of the vertex is 1. Substitute 1 for x.
= –16(1)2 + 32(1)
= –16(1) + 32
= –
Simplify.
= 16
The y-coordinate is 16.
The vertex is (1, 16).

11. Example 2 Continued
Step 3 Find the y-intercept.
f(x) = –16x2 + 32x + 0
Identify c.
The y-intercept is 0; the graph passes through (0, 0).

12. Example 2 Continued
Step 4 Graph the axis of symmetry, the vertex, and the point containing the y-intercept. Then reflect the point across the axis of symmetry. Connect the points with a smooth curve.
(0, 0)
(1, 16)
(2, 0)

13. Example 2 Continued
The vertex is (1, 16). So at 1 second, the basketball has reached its maximum height of 16 feet. The graph shows the zeros of the function are 0 and 2. At 0 seconds the basketball has not yet been thrown, and at 2 seconds it reaches the ground. The basketball is in the air for 2 seconds.
(0, 0)
(1, 16)
(2, 0)

14.   Example 2 Continued 4 Look Back
Check by substitution (1, 16) and (2, 0) into the function.
16 = –16(1)2 + 32(1) ?
16 = – ?
16 = 16 
0 = –16(2)2 + 32(2) ?
0 = – ?
0 = 0 

15. The vertex is the highest or lowest point on a parabola
The vertex is the highest or lowest point on a parabola. Therefore, in the example, it gives the maximum height of the basketball.
Remember!

16. Check It Out! Example 2
As Molly dives into her pool, her height in feet above the water can be modeled by the function f(x) = –16x2 + 16x + 12, where x is the time in seconds after she begins diving. Find the maximum height of her dive and the time it takes Molly to reach this height. Then find how long it takes her to reach the pool.

17. Check It Out! Example 2 Continued1
Understand the Problem
The answer includes three parts: the maximum height, the time to reach the maximum height, and the time to reach the pool.
List the important information:
The function f(x) = –16x2 + 16x + 12 models the height of the dive after x seconds.

18. Check It Out! Example 2 Continued
Make a Plan
Find the vertex of the graph because the maximum height of the dive and the time it takes to reach it are the coordinates of the vertex. The diver will hit the water when its height is 0, so find the zeros of the function. You can do this by graphing.

19. Check It Out! Example 2 Continued
Solve 3
Step 1 Find the axis of symmetry.
Use x = Substitute –16 for a and 16 for b.
Simplify.
The axis of symmetry is x = 0.5.

20. Check It Out! Example 2 Continued
Step 2 Find the vertex.
f(x) = –16x2 + 16x + 12
The x-coordinate of the vertex is 0.5.
Substitute 0.5 for x.
= –16(0.5)2 + 16(0.5) +12
= –16(0.25)
Simplify.
= –4 + 20
= 16
The y-coordinate is 9.
The vertex is (0.5, 16).

21. Check It Out! Example 2 Continued
Step 3 Find the y-intercept.
f(x) = –16x2 + 16x + 12
Identify c.
The y-intercept is 12; the graph passes through (0, 12).

22. Check It Out! Example 2 Continued
Step 4 Find another point on the same side of the axis of symmetry as the point containing the y-intercept.
Since the axis of symmetry is x = 0.5, choose an x-value that is less than 0.5.
Let x = 0.25
f(x) = –16(0.25)2 + 16(0.25) + 12
Substitute 0.25 for x.
= –
Simplify.
= 15
Another point is (0.25, 15).

23. Check It Out! Example 2 Continued
Step 5 Graph the axis of symmetry, the vertex, the point containing the y-intercept, and the other point. Then reflect the points across the axis of symmetry. Connect the points with a smooth curve.

24. Check It Out! Example 2 Continued
The vertex is (0.5, 16). So at 0.5 seconds, Molly's dive has reached its maximum height of 16 feet. The graph shows the zeros of the function are –0.5 and 1.5 seconds. At –0.5 seconds the dive has not begun, and at 1.5 seconds she reaches the pool. Molly reaches the pool in 1.5 seconds.

25.   Check It Out! Example 2 Continued 4 Look Back
Check by substituting (0.5, 16) and (1.5, 0) into the function.
16 = –16(0.5)2 + 16(0.5) + 12 ?
16 = – ?
16 = 16 
0 = –16(1.5)2 + 16(1.5) +12 ?
0 = – ?
0 = 0 

26. Check It Out! Example 2
What if…? A dolphin jumps out of the water. The quadratic function y = –16x x models the dolphin’s height above the water after x seconds. How long is the dolphin out of the water?
When the dolphin leaves the water, its height is 0, and when the dolphin reenters the water, its height is 0. So solve 0 = –16x2 + 32x to find the times when the dolphin leaves and reenters the water.
Step 1 Write the related function
0 = –16x2 + 32x
y = –16x2 + 32x

27. Check It Out! Example 2 Continued
Step 2 Graph the function.
Use a graphing calculator.
Step 3 Use to estimate the zeros.
The zeros appear to be 0 and 2.
The dolphin leaves the water at 0 seconds and reenters at 2 seconds.
The dolphin is out of the water for 2 seconds.

28. Example 3: Application
The height in feet of a diver above the water can be modeled by h(t) = –16t2 + 8t + 8, where t is time in seconds after the diver jumps off a platform. Find the time it takes for the diver to reach the water.
h = –16t2 + 8t + 8
The diver reaches the water when h = 0.
0 = –16t2 + 8t + 8
0 = –8(2t2 – t – 1)
Factor out the GFC, –8.
0 = –8(2t + 1)(t – 1)
Factor the trinomial.

29.   Example 3 Continued Use the Zero Product Property.
–8 ≠ 0, 2t + 1 = 0 or t – 1= 0
2t = –1 or t = 1
Solve each equation.
Since time cannot be negative, does not make sense in this situation. 
It takes the diver 1 second to reach the water.
Check 0 = –16t2 + 8t + 8
0 –16(1)2 + 8(1) + 8
0 –
Substitute 1 into the original equation. 

30. Lesson Quiz
2. The height in feet of a fireworks shell can be modeled by h(t) = –16t t, where t is the time in seconds after it is fired. Find the maximum height of the shell, the time it takes to reach its maximum height, and length of time the shell is in the air.
784 ft; 7 s; 14 s

31. Lesson Quiz: Part II
7. A rocket is shot straight up from the ground. The quadratic function f(t) = –16t2 + 96t models the rocket’s height above the ground after t seconds. How long does it take for the rocket to return to the ground?
6 s
8. The height of a rocket launched upward from a 160 foot cliff is modeled by the function h(t) = –16t2 + 48t + 160, where h is height in feet and t is time in seconds. Find the time it takes the rocket to reach the ground at the bottom of the cliff.
5 s