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Maximum and Minimum Values - Exercise Problems | MATH 1310, Study notes of Algebra

University of Houston (UH)Algebra

Material Type: Notes; Class: College Algebra; Subject: (Mathematics); University: University of Houston; Term: Unknown 2009;

Typology: Study notes

Pre 2010

Uploaded on 08/18/2009

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koofers-user-ast 🇺🇸

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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 khxaxf +−= 2
)()( .
(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. 76)( 2++= xxxf
2. 218)( 2+−= xxxf
3. xxxf 2)( 2−=
4. xxxf 10)( 2+=
5. 1182)( 2+−= xxxf
6. 15183)( 2++= xxxf
7. 98)( 2−−−= xxxf
8. 74)( 2−+−= xxxf
9. 115404)( 2+−= xxxf
10. 8105)( 2+−= xxxf
11. 1482)( 2−−−= xxxf
12. 27244)( 2−+−= xxxf
13. 35)( 2+−= xxxf
14. 17)( 2−+= xxxf
15. 2
432)( xxxf −−=
16. 2
37)( xxxf −−=
For each of the quadratic functions given below:
(a) Find the vertex ),( kh of the parabola by using
the formulas a
b
h2
−= and
(
)
a
b
fk 2
−= .
(b) State the maximum or minimum value of the
function, and state whether it is a maximum
or a minimum.
17. 5012)( 2+−= xxxf
18. 1014)( 2−+−= xxxf
19. 9162)( 2−+−= xxxf
20. 29123)( 2+−= xxxf
21. 13)( 2++= xxxf
22. 27)( 2+−= xxxf
23. 392)( 2++−= xxxf
24. 56)( 2−+−= xxxf
For each of the following problems, find a quadratic
function satisfying the given conditions.
25. Vertex )5,2(
−
; passes through )70,7(
26. Vertex )8,1(
−
−
; passes through )10,2(
27. Vertex )7,5( ; passes through )4,3(
28. Vertex )3,4(
−
; passes through )13,1(
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
tttH 8016)( 2+−= .
(a) Find the time at which the rocket reaches its
maximum height.
(b) Find the maximum height of the rocket.
32. A manufacturer has determined that their daily
profit in dollars from selling x machines is given
by the function
2
1.050200)( xxxP −+−= .
Using this model, what is the maximum daily
profit that the manufacturer can expect?

Partial preview of the text

Download Maximum and Minimum Values - Exercise Problems | MATH 1310 and more Study notes Algebra in PDF only on Docsity!

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 )= x^2 + 6 x + 7

2. f ( x )= x^2 − 8 x + 21

3. f ( x )= x^2 − 2 x

4. f^ (^ x )= x^2 +^10 x

5. f ( x )= 2 x^2 − 8 x + 11

6. f ( x )= 3 x^2 + 18 x + 15

7. f ( x )= − x^2 − 8 x − 9

8. f (^ x )=^ − x^2 +^4 x −^7

9. f ( x )= 4 x^2 − 40 x + 115

10. f ( x )= 5 x^2 − 10 x + 8

11. f ( x )= − 2 x^2 − 8 x − 14

12. f (^ x )=^ −^4 x^2 +^24 x −^27

13. f ( x )= x^2 − 5 x + 3

14. f ( x )= x^2 + 7 x − 1

15. f ( x )= 2 − 3 x − 4 x^2

16. f ( x )= 7 − x − 3 x^2

For each of the quadratic functions given below:

(a) Find the vertex ( h , k )of the parabola by using

the formulas h = − 2 ba and k = f ( − 2 ba ).

(b) State the maximum or minimum value of the

function, and state whether it is a maximum

or a minimum.

17. f ( x )= x^2 − 12 x + 50

18. f ( x )= − x^2 + 14 x − 10

19. f ( x )= − 2 x^2 + 16 x − 9

20. f ( x )= 3 x^2 − 12 x + 29

21. f ( x )= x^2 + 3 x + 1

22. f ( x )= x^2 − 7 x + 2

23. f ( x )= − 2 x^2 + 9 x + 3

24. f ( x )= − 6 x^2 + x − 5

For each of the following 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 )= − 16 t^2 + 80 t.

(a) Find the time at which the rocket reaches its

maximum height.

(b) Find the maximum height of the rocket.

32. A manufacturer has determined that their daily

profit in dollars from selling x machines is given

by the function

P ( x )= − 200 + 50 x − 0. 1 x^2.

Using this model, what is the maximum daily

profit that the manufacturer can expect?