1. Finding Maximums &
Minimums of
Polynomial Functions
fguilbert

2. Objective - find the maximum and minimum value for a function.
fguilbert

3. value of a quadratic is at the vertex.
The maximum or minimum
value of a quadratic is at
the vertex.
x = - b 2a
fguilbert

4. A rectangular dog pen is
Ex 1
A rectangular dog pen is
constructed with 60 m of fence.
One side of the pen is a barn.
What dimensions will give the
greatest area.
fguilbert

5. barn pen fence 60 m of fence. One side of the pen is a barn.
Find greatest area.
barn
pen
fence
fguilbert

6. Total of 3 sides is 60.
barn
pen
X X
60 - 2X
fguilbert

7. Area = x ( x)
= - 2x2 + 60x
barn
pen
X X
60 - 2X
fguilbert

8. Area = x ( x)
= - 2x2 + 60x
vertex (15, 450)
fguilbert

9. Area = x ( 60 - 2x) = - 2x2 + 60x (15, 450) barn pen 15 15 30
fguilbert

10. A box is made. Find x 10 6 6 x 10 sheet of metal.
ex 2
Squares with side x are cut from
corners.
A box is made. Find
x to maximize volume. 10 6 x
fguilbert

11. A cubic function has a local max and local min.
fguilbert

12. 10 6
x x
x x
x x
x x
fguilbert

13. Visualize cutting corners & folding the box.10 6
x x
x x
x x
x x
Visualize cutting corners
& folding the box.
fguilbert

14. 10 - 2x x x x x 6-2x x x x x Sides of the box are
shorter than 10 by 6.
fguilbert

15. bottom
6-2x
10 - 2x
fguilbert

16. x
bottom
6-2x
10 - 2x
What is the height ?
fguilbert

17. Volume = l w h x 6-2x 10 - 2x V = (10 - 2x )( 6 - 2x ) x bottom
fguilbert

18. V = (10 - 2x )( 6 - 2x ) x
max when x = 1.21
V = 32.8
fguilbert

19. Ex A ball is thrown up. Height is a function of time
Given : ho = 15m
vo= 20 m/sec
g = 9.8m/s2
h = height
fguilbert

20. h0 = 15 Vo =20 g =9.8 h = - 1/2 g t2 + vot + ho
h = - 1/2 (9.8) t t + 15
h = t t
fguilbert

21. b. Find ball height when t = 3
h = t t + 15
h = -4.9( 3)2 + 20(3) + 15
h = 30.9 meters
fguilbert

22. When is the ball 25 m off ground?
Ball is up 25M
so let h = 25
h = t t + 15
25 = t t
solve for t
fguilbert

23. 25 = - 4. 9 t2 + 20 t + 15 Put in standard form Get 0 on left. 0 = - 4
25 = t t Put in standard form Get 0 on left. 0 = t t Solve with quadratic formula.
fguilbert

24. t = √ 204
- 9.8
t = .6 or t = 3.5
fguilbert

25. When will ball hit ground? Looking for t? Or h??
when the ball hits ground
0 = t t
fguilbert

26. 0 = - 4.9 t2 + 20 t + 15 Use quadratic formula t = 4.68 or
t = -.6 (is this ok?) no
fguilbert

27. When will the ball be at the highest point? Find the maximum height.
h = t t + 15
fguilbert

28. I know how to get maximum height. Just graph the equation. Find the
y value of the max.
fguilbert

29. Find the Max
( t, h)
(2.04, 35.41)
h = t t +15
fguilbert

30. Effort - It is a solo performance that only you can control."
fguilbert