UMUC MATH-107 Final Exam Information
What should you know for the final exam ?
Here are some highlights of textbook material you should study in preparation for the final exam.
Review this material from the book and work all 30 problems on the attached practice exam. The
detailed instructor solutions are at the end of this package. The actual final exam is VERY close to
this practice exam.
Section
R.2
1.1
1.2
1.3
1.4
1.5
1.6
2.1
2.2
2.3
2.4
2.5
2.6
3.2
3.3
3.4
3.5
3.6
4.1
4.2
4.4
4.5
Material
Reducing fractions with exponents (powers)
Circle equations, Distance formula, Midpoint formula
Domain and Range of functions
Slope (given two points or an equation)
Equation of lines (from two points, or a point and a slope)
Decreasing and Increasing intervals
Composition functions
Linear equation word problems
Complex numbers
Quadratic functions
Parabola equations (finding vertex by completing the square)
Solving equations with sqaure roots and fractions
Solving inequality equations
“End Shape” of functions
Long division of polynomials
Zeros of a function
Asymptotes (vertical and horizontal), Domain of fractions
Solving inequality equations (using interval tests)
Inverse functions
Graphs of exponents and exponent word problems
Log Properties
Solving Exponent and Logarithm equations
MATH 107 PRACTICE EXAM
Closed Book, Calculator Permitted.
NOTE: Equation Sheet and Solutions at the End
MULTIPLE CHOICE
1. Find the center and radius of the circle with equation (x − 6 )2 + ( y + 15)2 = 4
A.
B.
C.
D.
(–6, 15); 4
(–6, 15); 2
(6, –15); 4
(6, –15); 2
2. The points (–2, 1) and (4, 3) are endpoints of the diameter of a circle. Find the length of the
radius of the circle.
A.
B.
C.
D.
2
2 2
10
2 10
3. Find the midpoint of the segment with endpoints (6, 1) and (– 4, 7).
A.
(–5, 3)
B.
(1, 4)
⎛7 3⎞
⎜ , ⎟
⎝2 2⎠
(5, –3)
C.
D.
2
(
3 ⎞
⎛
4
⎜ − 5a b 2 ⎟ a 7 b −1
⎠
4. Simplify: ⎝
5 2
a b
A.
− 10a 8 b
1
2
−3
B.
10a18 b
C.
25a10 b
D.
25a10
4
)
5. Which of the following equations does the graph represent?
A.
− 2x + y = 1
B.
− x + 2y = 2
C.
x − 2y = 2
D.
2x − y = 2
6. (7 pts) Suppose it costs $18.00 per day plus $0.25 per mile driven to rent a car.
If the total rental charge for one day is $100.50, how many miles were driven?
A.
B.
C.
D.
330
402
474
551
7. Find the slope of the line through the points (5, –6) and (2, –6).
12
7
A.
−
B.
C.
D.
0
4
Undefined
8. Write a slope-intercept equation for a line passing through the point (4, 1) and
parallel to the line y = 3x + 5.
A.
B.
C.
D.
y = 5x − 19
y = 3x + 1
y = 3x − 11
y = 3x + 11
For #9 and #10, consider the following graph
4
2
-4
2
-2
4
-2
-4
9. What are the domain and the range?
A.
B.
C.
D.
Domain [–4, 2] and Range [–2, 2]
Domain [–3, 1] and Range [–2, 2]
Domain [–3, 0] and Range [0, 2]
Domain [–2, 2] and Range [–4, 2]
10. On what interval is the function decreasing?
A.
B.
C.
D.
(– 4, –1)
(– 3, 0)
(– 2, 0)
(– 1, 1)
11. Which of the following functions has a graph which is symmetric with respect to y-axis?
A. f(x) = 6x – 2
B. f(x) = 1 – x2
C. f(x) = |2x + 4|
x
D. f ( x) =
x −3
12. Given f (x) = x2 – 3x + 5 and g(x) = x – 2, find the composite function ( f o g )( x) and simplify.
A.
B.
C.
D.
x2 – 7x + 15
x2 – 3x – 1
x2 – 3x + 9
x3 – 5x2 + 11x – 10
13. Simplify (8 + 3i)2 and write the answer in the form a + bi, where a and b are real numbers.
A.
B.
C.
D.
55
73
55 + 48i
73 + 48i
14. Find the exact solutions: 2 x 2 − x − 5 = 0 .
A.
–1, 5/2
1
39
−
i,
4
4
B.
1
39
+
i
4
4
C.
1
41
,
−
4
4
1
41
+
4
4
D.
None of the above
15. Find the vertex for the graph of the function f (x) = x2 + 6x + 7.
A.
B.
C.
D.
( 3, –2)
(–3, –2)
(–3, 16)
( 3, 16)
16. Solve:
A.
B.
C.
D.
2− x = x−2
–2, 2
1
1, 2
2
17. Given f ( x) = x − 5 and g ( x) = x − 7 , find the domain
of the quotient function f / g .
A.
B.
C.
D.
(− ∞, 5) ∪ (5, ∞ )
(− ∞, 7]
(− ∞, 7 ) ∪ (7, ∞ )
(7, ∞ )
18. Solve
A.
B.
C.
D.
x −1
x
=
x + 2 x +1
−
1
2
1
2
3
2
No solution
19. Which of the following describes the end behavior of the graph of
the function f(x) = 6x3 + 2x2 + 5x – 1 ?
A.
Upward to the left and upward to the right
vvvv
B.
vvvv
Downward to the left and downward to the right
C.
vvvv
Downward to the left and upward to the right
D.
vvvv
Upward to the left and downward to the right
20. Find the quotient and remainder when 4x3 – 3x2 – 9x – 5 is divided by x – 2.
A.
4x2 – 11x + 13, Remainder 67
B.
4x2 – 5x + 19, Remainder –33
C.
4x2 + 5x + 1, Remainder –3
D.
4x2 + 11x + 31, Remainder 67
21. Find all of the zeros of the polynomial 2x3 + 3x2 – 8x + 3.
A.
B.
C.
D.
1, 2, 3
1
–1, , 3
3
–3, 1
1
–3, , 1
2
22. Find all vertical asymptotes for the rational function f ( x) =
C.
1
2
9
x=
2
x = –3 and x = 3
D.
x = –1 and x = 1
A.
B.
2x 2 − 2
.
x=
23. Solve, and write the answer in interval notation:
A.
B.
C.
D.
x2 − 9
x(x – 4) ≥ 0
[0, ∞ )
[0, 4]
[4, ∞ )
(− ∞, 0] ∪ [4, ∞ )
24. Solve, and write the answer in set notation: |7 – 2x| < 9.
A.
B.
C.
D.
{x| −1 < x < 8}
{x| x < −1}
{x| x > 8}
{x| x < −1 or x > 8}
25. Given the function f ( x) = 3x + 5 , find a formula for the inverse function.
A.
f
B.
f
C.
f
D.
f
−1
(x ) = x − 3
5
−1
(x ) = x − 5
3
−1
(x ) =
−1
(x ) = 3x − 5
1
3x + 5
–x
26. Determine which is the graph of y = 2 + e .
A.
B.
C.
D.
(
)
27. The value of a stock is given by the function V (t ) = 20 + 58 1 − e −1.1t where V is the value of
the stock after time t, in months. Find the value of the stock after 2 months.
A.
B.
C.
D.
26.43
31.60
71.57
77.32
⎛ 1 ⎞
28. Solve: x = log 3 ⎜ ⎟
⎝ 27 ⎠
A. −9
B. −3
1
C.
3
1
D.
9
29.Express as a single logarithm:
⎛ 8 p⎞
⎟
A. log⎜
⎜ q ⎟
⎝
⎠
B.
⎛8+ p ⎞
⎟
log⎜
⎜ q ⎟
⎝
⎠
C.
log 8 +
D.
log (− 3 pq )
(
p −q
30. Solve 4 3 x −5 = 16 .
A.
1
3
B.
11
6
C.
7
3
D.
3
)
3 log 2 + ½ log p – log q.
MATH 107 FORMULAS
Properties of exponents
a pa q = a p+q
a p/a q = a p–q
(a p)q = a p q
(ab) p = a p b p
Factoring
(a + b)2 = a2 + 2ab + b2
a2 – b2 = (a + b)(a – b)
a3 + b3 = (a + b)(a2 – ab + b2)
(a – b)2 = a2 – 2ab + b2
a3 – b3 = (a – b)(a2 + ab + b2)
⎧ x for x ≥ 0
⎩− x for x < 0
Absolute value: | x |= ⎨
Distance
The distance between a and b on the number line is | a − b | .
The distance between points (x1, y1) and (x2, y2) is given by
(x2 − x1 )2 + ( y 2 − y1 )2 .
Midpoint Formula: If the endpoints of a line segment are (x1, y1) and (x2, y2) then the coordinates of the
⎛ x1 + x 2 y1 + y 2 ⎞
,
⎟
2 ⎠
⎝ 2
midpoint are ⎜
Equation of a Circle:
The equation of a circle with center (h, k) and radius r, in standard form, is (x − h ) + ( y − k ) = r 2
2
2
Slope
y − y1
.
The slope of a line containing points (x1, y1) and (x2, y2) where x1 ≠ x2, is given by m = 2
x2 − x1
Slope-Intercept Equation: y = mx + b, where m is the slope and (0, b) is the y-intercept.
Point-Slope Equation: y – y1 = m(x – x1), where m is the slope and (x1, y1) is a point on the line.
Composition of functions: ( f o g )( x ) = f ( g ( x) ) where x is in the domain of g and g(x) is in the domain of f.
Symmetry of functions
Even function ⇔ Graph symmetric with respect to y-axis ⇔ f(x) = f(–x) for all x in the domain
Odd function ⇔ Graph symmetric with respect to origin ⇔ f(x) = – f(–x) for all x in the domain
Transformation of function y = f(x)
Vertical translation:
Graph of y = f(x) + b is the graph of y = f(x) shifted upward b units
Graph of y = f(x) – b is the graph of y = f(x) shifted downward b units
Horizontal translation
Graph of y = f(x – d) is the graph of y = f(x) shifted rightward d units
Graph of y = f(x + d) is the graph of y = f(x) shifted leftward d units
Reflection
Graph of y = – f(x) is the reflection of the graph of y = f(x) across the x-axis
Graph of y = f(–x) is the reflection of the graph of y = f(x) across the y-axis
Vertical Stretching or Shrinking:
Graph of y = af(x) is a vertical stretching/shrinking of the graph of y = f(x)
Horizontal Stretching or Shrinking
Graph of y = f(cx) is a horizontal stretching/shrinking of the graph of y = f(x)
Complex number: a + bi, where a and b are real numbers and i =
Quadratic Formula
x=
−1
− b ± b 2 − 4ac
2a
Quadratic Functions f(x) = ax2 + bx + c
f(x) = a(x – h)2 + k
h=−
b
⎛ b ⎞
, k = f ⎜−
⎟ = f ( h)
2a
⎝ 2a ⎠
Absolute Value Inequalities
For a > 0, |x| < a ⇔ –a < x < a
For a > 0, |x| > a ⇔ x < –a or x > a
Polynomials
n-1
n
Polynomial Function: P(x) = an x + an-1 x
+ an-2 xn-2 + … + a1x + a0
Intermediate Value Theorem: For any polynomial P(x) with real coefficients, if P(a) and P(b) have opposite signs,
then the function P must a real zero between a and b.
Remainder Theorem: Given polynomial f(x) and value c, when dividing f(x) by x – c, the
remainder is f(c). That is, f(x) = (x – c) Q(x) + R, where R = f(c).
Factor Theorem: For a polynomial f(x), if f(c) = 0, then x – c is a factor of f(x).
Fundamental Theorem of Algebra: Every polynomial function of degree n, with n ≥ 1, has at least one zero in the
system of complex numbers.
Polynomial with real coefficients For a nonzero, a + bi is a zero ⇔ a – bi is a zero.
For a and c rational and b not a perfect square,
Polynomial with rational coefficients
a + c b is a zero ⇔ a − c b is a zero.
Polynomial with integer coefficients
Rational Zeros Theorem: If the coefficients are all integers, then for any rational zero (which is a zero of the form
p/q) the numerator p is a factor of the constant term a0 and q is a factor of the leading coefficient an.
Descartes’ Rule of Signs The number of positive real zeros of P(x) is either the same as the number of variations
of sign in P(x) or less than the number of variations of sign in P(x) by a positive even integer. The number of negative
real zeros of P(x) is either the same as the number of variations of sign in P(–x) or less than the number of variations of
sign in P(–x) by a positive even integer. A zero of multiplicity m must be counted m times.
Rational Function: A quotient of two polynomials, f ( x) =
p ( x)
, where p(x) and q(x) are polynomials and
q ( x)
q(x) is not the zero polynomial.
Direct Variation
Inverse Variation
y = kx, where k is a positive constant
k
y = , where k is a positive constant
x
Exponential Function with base a: f(x) = ax, where x is a real number, a > 0 and a ≠ 1.
Logarithms
y = loga x ⇔ x = a y
loga (MN) = loga M + loga N
loga (M p) = p loga M
loga 1 = 0
⎛M
log a ⎜
⎝N
loga a = 1
⎞
⎟ = log a M − log a N
⎠
loga ax = x
SOLUTIONS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
D
C
B
D
B
A
B
C
A
A
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
B
A
C
C
B
D
D
A
C
C
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
D
D
D
A
B
B
C
B
A
C