Math 95 Practice Exam 2 - Portland Community College

Math 95 Practice Exam 2

Part 1: No Calculator

Show all your work so that:

? someone who wanted to know how you found your answer can clearly see how. ? if you make a mistake, I can see where it happened and determine how much partial credit

you should be awarded.

You may use scratch paper, but all necessary work must be written on this exam. Simplify all fractions as much as possible. The entire exam is closed-note, closed-book. You may not use your calculator or any other electronic device on this part of the exam. Take your time, because you have plenty to spare. Check your answers .

1. Use the graph of the quadratic function shown in Figure 1 to complete the following. Approximating any values with a decimal that is reasonably close is OK.

Figure 1. Graph of y = f (x)

a) State the vertex as an ordered pair.

b) Write the vertex form for the function f .

(Use

a

=

-

4 9

.)

c) What is the domain of f ?

d) What is the range of f ?

e) Solve f (x) = 3 in set notation.

f) Solve f (x) 0 in interval notation.

y 5

-4

-2

x

2

4

6

-5

2. Use the graph of the quadratic function shown in Figure 2 to complete the following. Approxi-

mating any values with a decimal that is reasonably close is OK. Figure 2. Graph of y = g(x)

a) State the x-intercept(s). b) State the y-intercept.

10 y

c) Write the factored form for the function g.

5

(Use a = 1.)

d) What is the domain of g? e) What is the range of g?

x

-10 -8

-6

-4

-2

2

f) Solve g(x) = -10 in set notation.

-5

g) Solve g(x) < -5 in interval notation.

-10

1

3. Let H(x) = -1.8(x - 9)2 + 2.

Math 95 Practice Exam 2

a) What is the vertex? c) State the domain in interval notation.

b) Does this graph open upward or downward? d) State the range in interval notation.

4.

Let

H (x)

=

3 2

(x

+

12)2

- 7.

a) What is the vertex? c) State the domain in interval notation.

b) Does this graph open upward or downward? d) State the range in interval notation.

5. Solve the quadratic equation Q2 - 8Q + 3 = 0 by completing the square. State the solution in set notation.

6.

Solve

the

quadratic

equation

x2

+

3 2

x

-

7 4

=

0

by

a

method

of

your

choice.

State

the

solution

in

set notation.

7. a) Let p(x) = (x - 9)(x - 1). Find the key

b) Let p(x) = -2(x + 1)2 + 8. Find the key

features of the graph of this function (x- features of the graph of this function (x-

intercepts, y-intercept, and vertex) and sketch intercepts, y-intercept, and vertex) and sketch

the graph of y = p(x) in Figure 3.

the graph of y = p(x) in Figure 4.

Figure 3. Graph of y = p(x)

Figure 4. Graph of y = p(x)

8. Solve each quadratic equation using the quadratic formula. State the solution in set notation. Use the imaginary unit i if the answer is complex. Then use your work to determine the number of x-intercepts each graph would have.

a) x2 + 1 = 3x

Part 1: No Calculator

b) 2x2 + 6x + 5 = 0

Page 2 of 7

Math 95 Practice Exam 2

9. Simplify each of the following expressions and state each complex number in standard form.

a) (1 + 3i) + (-2 + i)

b) (-5 - 6i) + (4 + 5i)

c) (1 + 3i) - (-2 + i)

d) (-5 - 6i) - (4 + 5i) g) (-5 - 6i)(4 + 5i)

-5 - 6i j)

4 + 5i

e) 3i(-2 + i) h) -5 - 6i

5i

f) (1 + 3i)(-2 + i) 1 + 3i

i) -2 + i

10. Use the graph of y = B(x) in Figure 5 to answer the following. Reasonable decimal approxi-

mations are acceptable.

Figure 5. Graph of y = B(x)

a) Find B(3) in function notation. b) Solve B(x) = 2 in set notation. c) Solve B(x) = -2 in set notation. d) Solve B(x) 2 in interval notation. e) State the domain of B using interval notation. f) State the range of B using interval notation.

y 6

4

2 -6 -4 -2

x

2

4

6

-2

11. Use the graph of y = C(x) in Figure 6 to answer the following. Reasonable decimal approxi-

mations are acceptable.

Figure 6. Graph of y = C(x)

y 6

a) Find C(2) in function notation. b) Solve C(x) = 4 in set notation. c) Solve C(x) > 4 in interval notation. d) State the domain of C using interval notation. e) State the range of C using interval notation.

4

2 y=1

x

-6 -4 -2

2

4

6

-2

-4

-6

x=4

Part 1: No Calculator

Page 3 of 7

Math 95 Practice Exam 2

x2 - 64

12. Let H(x) =

.

x2 + 17x + 72

a) Find and state the domain of H using set builder notation.

b) Simplify H(x), making sure to state any restrictions.

2x2 + 3x - 5

13. Let v(x) =

.

x2 + 17x - 18

a) Find and state the domain of v using set builder notation.

b) Simplify v(x), making sure to state any restrictions.

14. Simplify each expression. If applicable, include any restrictions.

x2 - 7x - 18 x + 6

a)

?

x+2 x-9

x2 + 3x + 2

x+2

c) x + 4 ? x2 + 9x + 20

Q2 + 13Q 5Q + 25 b) Q3 - 25Q ? Q + 13

Z +3

Z2 + 3Z

d) Z2 + 13Z + 30 ? Z2 - 100

15. Add or subtract. If applicable, include any restrictions.

x+3 5

a)

+

x-5 x-3

x

4

c) x - 1 - x2 + 2x - 3

x x-1 b) 2x - 4 - x - 2

16. Simplify the complex fractions. If applicable, include any restrictions.

2

2+

a)

3 1

2- 4

12

+

c)

x-1 x 1

2- x

2x + 3

b)

x x-4

x

17. Solve and check. Write answers in set notation. Explain any extraneous solutions.

x

4

a)

=

x+2 x-3

4

x

18

c)

+ x+3 x-3

=

x2 - 9

1

1

b) 3x - 2 = 2x

Part 1: No Calculator

Page 4 of 7

Name:

Part 2: Calculator Permitted

You may use a calculator (basic, scientific, or graphing), but may not use any other electronic device. Show all your work so that:

? someone who wanted to know how you found your answer can clearly see how. ? if you make a mistake, I can see where it happened and determine how much partial credit

you should be awarded. The calculator should only be used at the end of your problem-solving process, to calculate some decimal value. Round where appropriate.

18. A microbe colony has a population (measured in thousands of individuals) that can be modeled

t3 - t + 1

by P (t) =

, where t is in days since the microbes first colonized the location. Find and

t2 + t + 2

interpret P (2.5). Use function notation in your answer.

19. A patient ingests a pill, and the concentration of the drug in that person's bloodstream (in 15t

mg/mL) after t hours is given by C(t) = t4 + t + 1 . Find and interpret C(0.5). Use function notation in your answer.

20. An object is thrown upward. It's height above ground level, h(t) (in meters), after t seconds can be modeled by h(t) = -4.9t2 + 31.4t + 6.5.

a) What is the maximum height? When does this occur? b) When does the object hit the ground? c) Solve and interpret h(t) = 40. d) Solve and interpret h(t) = 80.

21. Suppose that a person with a push mower can mow a large lawn in 5 hours, whereas the same lawn can be mowed with a riding mower in 2 hours.

a) Write an equation whose solution gives the time needed to mow the lawn if both mowers are used at the same time.

b) Solve the equation in part a.

Part 2: Calculator Permitted

Page 5 of 7

Answers

1. a) (2, 4) d) (-, 4]

b)

f (x)

=

-

4 9

(x

-

2)2

+

4

e) {0.5, 3.5}

c) (-, ) f) (-, -1] [5, )

2. a) (-7, 0) and (-1, 0) d) (-, ) g) (-6, -2)

b) (0, 7). e) [-9, )

c) g(x) = (x + 7)(x + 1) f)

3. a) (9, 2)

b) downward

c) (-, )

d) (-, 2]

4. a) (-12, -7)

b) upward

5. The solution set is {4 + 13, 4 - 13}.

c) (-, )

d) [-7, )

{

}

6. The solution set is

-

3 4

+

37 4

,

-

3 4

-

37 4

.

7.

y

10 (0, 9)

(1, 0)

(9, 0x)

-4 -2

2 4 6 8 10 12

10

(-1, 8)

5

y

(0, 6)

(-3, 0)

-10

-5

(1, 0)

x

5

10

-10

-5

(5, -16) a)

-10

b)

8. a)

The

solution

set

is

{

3- 2

5,

3+ 2

5 }.

The discriminant is greater than 0 so there are two x-

intercepts. They are approximately (.38, 0) and (2.6, 0)

b)

The

solution

set

is

{

-3-i 2

,

-3+i 2

}.

The

discriminant

is

less

than

0

so

the

solutions

are

complex

and cannot be graphed. This means that there are no x-intercepts.

9. a) -1 + 4i f) -5 - 5i

b) -1 - i g) 10 - 49i

c) 3 + 2i h) -6 + 5i

5

d) -9 - 11i i) 1 - 7i

5

e) -3 - 6i j) -50 + i

41

10.a) B(3) 1.25 b) x = -1 or x = 2

11. a) C(2) -1.2.

c) This equation has no solutions.

b) x 5.5.

d) [-1, 2]

c) About (4, 5.5).

e) [-3, )

d) (-, 4) (4, )

f) About [0, 6.2]

e) (-, 1) (1, )

12.a) {x|x = -9, x = -8}

b)

H (x)

=

x

-

8 ,

where

x

=

-8

x+9

Answers

13.a) {x|x = -18, x = 1}

2x + 5

b) v(x) =

, where x = 1

x + 18

Page 6 of 7

14.a) x + 6, where x = -2 and x = 9

c) (x + 1)(x + 5), where x = -4 and x = -2

15.a) x2 + 5x - 34 (x - 5)(x - 3)

x+4

c)

, where x = 1

x+3

16.a)

32 21

c)

3x - 2

(2x - 1)(x - 1)

17.a) {-1, 8}

c) {-10} The answer 3 is an extraneous solution because it makes a denominator equal to 0.

5

b)

, where Q = -13 and Q = -5 and

Q-5

Q = 0

d) Z - 10 , where Z = -10 Z(Z + 3)

b)

-

1 2

,

where

x

=

2

2x + 3

b)

, where x = 0

x-4

1 b) {- 12}

18. P (2.5) 1.31. After 2.5 days, there are about 1.31 thousand microbes (about 1310).

19. C(0.5) = 4.8. After half an hour, the drug concentration in the patient's blood is 4.8 mg/mL.

20.a) After about 3.204 seconds, the object reaches its maximum height of 56.8 meters. b) The object hit the ground after about 6.61 seconds.

c) t 1.35 and t 5.06. After about 1.35 seconds the object reaches a height of 40 meters in the air. Later, at about 5.06 seconds after launch, the object returns to being 40 meters high

on its way down.

d) This equation has no solutions. The object never reaches 80 meters high.

21.a)

1 +

1

=

1

52 x

b) It would take about 1.43 hours to mow the lawn if both mowers are used.

Answers

Page 7 of 7

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