Mathematics 120 (Precalculus) (Fall, 2006)

Review: Final Examination

prepared by:

Joseph Malkevitch
Mathematics Department
York College (CUNY)
Jamaica, NY 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

1. Determine the value(s) of x for which the following equations or inequalities are satisfied.

a. x = 12

b. -x = 10

c. 2x = 18

d. -9x = 36

e. -x = 14

f. -5x = -35

g. 2x -8 < x + 5

h. x + 11 = -2

i. 2x - 13 = 7

j. 2x + 16 = -4 + 3x

k. 5(x - 4) = 2x - 40

l. -2x > 12

m. 2x -8 < x + 5

n. -x + 3 > -2x - 12

o. -2(x - 9) > 4x +6

p. -(x - 9) < x -17

q. 3x - x2 = 5x -22 - x2

r. x2 - 6x ³ 0

s. -x2 - 5x + 15 > 0

2. Sketch the graph of the following equations:

a. x = 6

b. y = -2

c. x + 2 y = 10

d. y = -x2 + 6

e. x2 + y2 = 25

f. x2 + y2 - 10x = 0

g. x2 + y2 + 6y - 4x = -2

h. (x + 4)2 + (y - 6)2 = (1/4)2

i. y = - | 2x |

j. y = | 3x - 6 |

k. 2x2 + 12 x + 2y2 - 8y = 10

l. x2 + 4x - 6 = 0

3. Sketch a graph of the functions: h(x) = -x2

Use the graph above to graph:

a. -h(x)

b. h(x) + 5

c. h(x-6)

d. 3h(x)

e. h(-x)

4. Write down using inequalities the points which are represented by the following intervals:

a. (-2, 3)

b. (-8, -2 ]

c. [ 4, 8 ]

d. (-°, 3)

e. (-2, ° )

5. Find the center and radius of the following circles and be prepared to draw a sketch of the circle which shows as least 4 points on the circle.

a. x2 + y2 = 64

b. x2 + y2 + 4x - 8y = -4

c. x2 - 10y + y2 + 6x = +2

d. x2 - 2y2 + 10y = -3y2

6. i. Given the function defined by the formula y = g(x) = -; f(x) = 2x - 3

a. g(5)

b. g(-2)

c. g(0)

d. g(-1/2)

e. g(a)

f. g(a +1)

g. f(x +h) - f(x)

h. Compute the x and y intercepts for the graph of y = f(x)

ii. What are the domain and range of f(x).and g(x)?

iii. Compute f º g (-4) and g º f (-1); (small circle means composition of the two functions). Compute f º g (x). Find inverse function of f(x).

7. Sketch the location of the points (-2, 3), (-3, 2), (4,2), (-4, -5).

8. Given triangle O = (0, 0), A = (3, 4) and B = (-5, -12), find the lengths of the sides of the triangle and its perimeter.

9. Are the functions g(x) = | -(x - 4) |, domain all real numbers and h(x) = | -(4 -x) | domain all real numbers the same function?

10. Write down the equation of the lines through the given points and indicate the slope of the lines.

a. (2, -4) (0, 5)

b. (-3, -2) (0, 6)

c. (-3, 4) (3, 7)

d. (-2, -3) (4, -7)

e. ( -1, 3) (-6, 3)

f. (2, 5) (2, -5)

11. For each of the equations below, determine the slope of the line and draw a sketch of the line.

a. 2x - 5y = 10

b. 3y + 5x = 15

c. -y = 4x + 8

d. 6x = 2y - 4

e. x = -3

f. y = -7/2

12. Given A = (2, -3) B= (0, 0) and C = (1, 7)

a. Find the lengths of the segments AB, BC, and CA

b. Find the equations of AB, BC, CA

c. Find the equation of a line through C parallel to the line y =4.

d. Find the equation of a line through B parallel to AC

e. Find the equation of a line through C perpendicular to AB.

f. Find the equation of a line through A perpendicular to the line x = 9

13. Compute the value of:

a.

b.

c.

d.

e.

f.

14. Determine if (cos(x))(1 + tan2 x) = sec x.

15. Use synthetic division to check:

a. Is -1 a root of x4 - x3 - 6x2 - x + 6 = 0? (Use synthetic division.)

b. The quotient when x4 - x3 - 6x2 - x + 6 is divided by x - 3.

c. The remainder when x4 - x3 - 6x2 - x + 6 is divided by x - 3.

d. Is x + 2 a factor of x4 - x3 - 6x2 + 6. (Hint: note there is no x term)

16. Sketch a graph of the following:

(Show where the graph cuts the x-axis, y-axis (if it does), (if they exist), (if there is one), behavior for large x and and large negative x):

a.

b.

c.

17. Simplify the following expressions using the laws of logarithms:

a.

b.

c.

18. i. Convert from degrees to radians:

a. -30 degrees
b. 180 degrees
c. 45 degrees
d. 135 degrees
e. 90 degrees
f. -180 degrees
g. 270 degrees
h. 210 degrees
i. -300 degrees
j. 595 degrees

ii. Compute the six trig functions of the angles above.

19. i. Convert from radians to degrees:

c. 13¹/3
d. -11¹/3
e. ¹/2

ii. Compute the six trig functions of the angles above.

20. Compute the sin t, cos t and tan t for the situations below:

a. t is quadrant I and sec t = 2

b. t is in quadrant III and cot t = 11/13

c. -t is is in quadrant II and csc -t = 3/4

d. t is in quadrant IV and sec t = 5/(Ã2)

e. t is in quadrant II and csc t = 7/(Ã5)

f. t is in quadrant I and sin t = 4/7

g. t is in quadrant II and cos t = -4/11

21. Draw a graph of y = cos t and y = sin t for 0 ² t ² 2¹.

22. Compute:

a. arc sin 1/2

b. arc tan 1

c. arc cos (-Ã3/2)

23. Graph y = 2 + sin t

24. Graph y = -3 cos t

25. Compute and write the answer in the form a + bi:

a. (2 + 4i)2 =

b. (3 + 4i)(-1 + 3i) =

c. (3 - 4i)-(7 + 2i) =

d. (-2 - 5i)2 =

e. ((2+5i)/(-1+2i))=

f. ((-2 + 4i)/(3 - 2 i)) =

g. (-3i)3 - (7i)2 - 3i =

26. Graph y = 2x

27. Graph y = -4(5x)

28. Know the trig functions for the 30-60-90 and 45-45-90 triangles.

29. Find the inverse function for:

a. y = 5x - 7

b. y = x1/3 + 11

c. y = (2x - 5)/(x + 2)

d. y = 5/x

e. y = 5/(x-11)

f. y = (3x + 4) - 7