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 - 3a. 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:
a. ¹/6 radians
b. 21¹ radians
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