Example 6:
Solution to Example 6:
Example 7:
Solution to Example 7:
Example 8:
Solution to Example 8:
Exercise Set 3.5: Maximum and Minimum Values
For each of the quadratic functions given below:
(a) Complete the square to write the equation in
the standard form f (x) = a(x − h)2 + k .
(b) State the coordinates of the vertex of the
parabola .
(c) Sketch the graph of the parabola .
(d) State the maximum or minimum value of the
function , and state whether it is a maximum
or a minimum.
1. f (x) = x2 + 6x + 7
2. f (x) = x2 − 8x + 21
3. f (x) = x2 − 2x
4. f (x) = x2 +10x
5. f (x) = 2x2 −8x +11
6. f (x) = 3x2 +18x +15
7. f (x) = −x2 − 8x − 9
8. f (x) = −x2 + 4x − 7
9. f (x) = 4x2 − 40x +115
10. f (x) = 5x2 −10x + 8
11. f (x) = −2x2 −8x −14
12. f (x) = −4x2 + 24x − 27
13. f (x) = x2 − 5x + 3
14. f (x) = x2 + 7x −1
15. f (x) = 2 − 3x − 4x2
16. f (x) = 7 − x − 3x2
For each of the quadratic functions given below:
(a) Find the vertex (h, k) of the parabola by using
the formulas
and
.
(b) State the maximum or minimum value of the
function, and state whether it is a maximum
or a minimum.
17. f (x) = x2 −12x + 50
18. f (x) = −x2 +14x −10
19. f (x) = −2x2 +16x − 9
20. f (x) = 3x2 −12x + 29
21. f (x) = x2 + 3x +1
22. f (x) = x2 − 7x + 2
23. f (x) = −2x2 + 9x + 3
24. f (x) = −6x2 + x − 5
For each of the fol lowing problems , find a quadratic
function satisfying the given conditions.
25. Vertex (2, − 5) ; passes through (7, 70)
26. Vertex (−1, −8) ; passes through (2,10)
27. Vertex (5, 7) ; passes through (3, 4)
28. Vertex (−4, 3) ; passes through (1,13)
Answer the following.
29. Two numbers have a sum of 10. Find the largest
possible value of their product .
30. Jim is beginning to create a garden in his back
yard. He has 60 feet of fence to enclose the
rectangular garden, and he wants to maximize
the area of the garden. Find the dimensions Jim
should use for the length and width of the
garden. Then state the area of the garden.
31. A rocket is fired directly upwards with a velocity
of 80 ft/sec. The equation for its height, H, as a
function of time, t, is given by the function
H(t) = −16t2 + 80t .
(a) Find the time at which the rocket reaches its
maximum height.
(b) Find the maximum height of the rocket.
32. A manufacturer has de termined that their daily
profit in dollars from selling x machines is given
by the function
P(x) = −200 + 50x − 0.1x2 .
Using this model, what is the maximum daily
profit that the manufacturer can expect?
