Name: ______________________
Student ID: _______________
MATH 1050-90
Practice Final
The purpose of the practice exam is to give you an idea of the following:
 length of exam
 difficulty level of problems
 your instructor’s problem writing style
Your actual exam will have different problems. You should review your homework, quizzes, lecture
notes etc., not just perfect taking the practice exam. However, you can use the practice exam to gauge
how well you know the material. To do this, take it under the same conditions as a normal exam (no
notes, no calculator, time yourself). Then score your problems against the solution key.
The following instructions are on the exam:
o
Use a PENCIL, erase or cross out errors.
o
SHOW ALL WORK. No points will be given for answers without justification.
o
Circle your answer so it is easy to locate.
o
NO CALCULATORS, NOTES, PHONES, ETC.
o
Answers should be simplified (reduced).
o
The value of each question is shown.
o
Finish in 140 minutes (two hours + 20
minute grace period). You are
responsible for keeping track of the
time. The proctor does not say “time
is up”. For each minute you take
beyond 140 min, your score will be
reduced by 0.5%.
Page 1
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TOTAL:
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Page 5
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/149
/100
Formulas:
Parent Functions
Parabola:
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
Square root
𝑦 = 𝑎√𝑥 − ℎ + 𝑘
Cubic polynomial
𝑦 = 𝑎(𝑥 − ℎ)3 + 𝑘
Rational
𝑎𝑥±𝑏
𝑦=
Exponential
𝑦 = ±𝑏 (𝑥−ℎ) + 𝑘
Logarithmic
𝑦 = ± log 𝑏 (ℎ − 𝑥) + 𝑘
𝑐𝑥±𝑑


y  Pe rt and
r
P(1 ) nt
n

Remember log(a) means log in base 10, and
ln(a) means base e.

The sum of
n terms of an arithmetic series:
Sn 

y=
n
(a  a )
2 1 n

The area of a triangle with vertices (a,b), (c,d),
𝑎 𝑏 1
1
and (e,f)
± | 𝑐 𝑑 1|
2
𝑒 𝑓 1

The sum of n terms of a geometric series:
Sn  a1

Absolute value
𝑦 = 𝑎|𝑥 − ℎ| + 𝑘
r n 1
r 1
Page 1 of 9
1. Lines. (9 pts)
Given points A = (2,-8) and B = (-3,-5) and C = (-4,6), find:
a. The distance between
A and B:
b. The slope of the line
containing A and B:
c. The equation of the line
passing through C and parallel
to the line containing A and B.
(in slope-intercept form, y=mx+b)
2. Calculations. (6 pts) 𝑓(𝑥) = 15x  20
5
g (x)  𝑥 − 2
b. g 1 ( x)
a. ( g  f )(10)
3. More Calculations. Calculate and simplify. (9 pts)
a. (1  3i)(1  3i)   16  i 7
b. log 5000 − log 5
c.
4
∑(−1)𝑛 𝑛3
𝑛=1
Page 2 of 9
4. Answering Questions about graphs. (16pts)
i(x)
(Hint: parent functions are on the front page)
a.) Circle the point(s) on the graph where i(x)= 5
b.) The approximate value(s) of x where i(x)=5 are:
f.) Which function(s) have asymptotes?
(circle all that apply)
______________
c.) i(x) is a rational function. Circle a possible
denominator of i(x).
0
3x+3
3x
3x – 3
d.) domain of i(x): ___________________
e.) range of i(x): _____________________
5. Graphing. (8 pts)
Given the function 𝑓(𝑥) = 3𝑥 2 − 6𝑥 − 2
a. Complete the square to rewrite it in the form
f(x) = a(x-h)2 +k.
f(x)
g(x)
h(x)
g.) f(x) = ___________________________
h.) g(x) = ___________________________
i.) h(x)= ___________________________
b. Graph the function using the work in a.
(Note: if you made a mistake in part a, you’ll get full credit for
part b if your graph matches your result in part a. If you
couldn’t do part a, state a function in the form f(x) = a(x-h)2 +k
and graph it.)
Page 3 of 9
6. Graphing. Be accurate. (9 pts)
Graph 𝑓(𝑥) = 4𝑥
Graph 𝑔(𝑥) = 4−𝑥 − 2
Graph the INVERSE of f(x)
7. Polynomials. (12 pts) You are given the polynomial 𝑝(𝑥) = 𝑥 4 − 2𝑥 3 + 9𝑥 2 − 18𝑥
a. Circle the picture that corresponds to the
possible shape of p(x).
d. Use the rational zero theorem to cross out values
of x which of these are NOT possible zeros.
1
± , ± 1,
2
±2,
±5,
±18
e. x=2 is a zero of p(x). Find all the zeros of
b. How many zeros of p(x) are there? _______
c. Show that x=3 is NOT a zero using
long/synthetic division.
p(x).
Page 4 of 9
8. Exponential and Logarithmic Problems. (8 pts) Simplify if possible.
a. Solve for x. If there are false
solutions, please note that they are
false.
log 3 (𝑥 + 1)2 = 2
b. Scientist monitoring a rainforest in Central America
estimate that it is decreasing at a rate of 2% each
year. They create a model:
𝑦 = 𝑃𝑒 −0.02𝑡 ,
where P is the area of the rain forest at the start of
2014 and y is the area after t years. Based on this
model, after how many years will the rain forest be
half the size it was at the start of 2014? Your
answer should be exact.
9. Matrix Operations (8 pts)
1 −2
2
A= [ 4
B=[ ]
C= [4 1]
0]
−6
−3 5
−1 3
D =[
]
4 −6
1
E=[ ]
−5
Circle the expression that is defined and calculate it:
a. AD
DA
b.
3C-B
3E-B
Page 5 of 9
10. Systems of Equations and Matrices (12 pts)
a. Write the matrix in row echelon form*.
2
[3
1
2 1
4
1 −2 5 ]
0 −3 −2
b. Solve the system of equations using your
work in part a.
If you didn’t do part a or you do not know how
to apply it, solve this system any way you can
for partial credit.
2𝑥 + 2𝑦 + 𝑧 = 4
{3𝑥 + 𝑦 − 2𝑧 = 5
𝑥 − 3𝑧 = −2
* Row echelon form:
1 # # #
[0 1 # #]
0 0 1 #
Page 6 of 9
11. The Binomial Theorem. (8 pts)
a.) Write Pascal’s triangle until the row
beginning with “1 7 …” .
b.) Find (a+b)6.
c.) What is the coefficient of a3b3 in the
expansion of (a+b)6?
d.) Find the coefficient of 𝑥 6 𝑦 3 in the
expansion of (𝑥 2 − 2𝑦)6 .
12. Using matrices to solve equations. (9 pts)
𝑎𝑥 + 𝑏𝑦 = 𝑐
𝑎
The system of equations {
can be modelled with the matrix equation [
𝑑𝑥 + 𝑒𝑦 = 𝑓
𝑑
𝑥
𝑎 𝑏 −1 𝑐
A solution to this equation is [𝑦 ] = [
] [𝑓 ].
𝑑 𝑒
a. Write the system of equations
4𝑥 + 6𝑦 = 1
{
−2𝑥 + 2𝑦 = 5
as a matrix equation.
b. Find the inverse for the 2x2 matrix in a.
𝑐
𝑏 𝑥
] ∙ [ ] = [𝑓 ].
𝑒 𝑦
c. Solve the system of equations using
matrix multiplication.
Page 7 of 9
13. Sequences and Series (7 pts)
a. What are the next two terms in the
9 9
9
sequence: − 8 , 4 , − 2, …
b. Is the sequence arithmetic, geometric or
neither?
c. If arithmetic, find the common difference. If
geometric, find the common ratio. If neither,
state this.
14. More Sequences and Series (8 pts)
Large quantities of water are measured using
the unit “acre foot”, which is the amount of
water is takes to cover one acre with one foot
of water.
The City is trying to encourage water
conservation by increasing the price of water
as consumers use more water.
$130 for the first acre foot
$170 for the second acre foot
$210 for the third acre foot
Assume the pattern continues.
a. Is the sequence involved arithmetic,
geometric or neither?
d. Write the formula for the nth term.
b. Write a formula for how much the nth acre
foot costs:
c. How much will the 20th acre-foot of water
cost?
e. Which term is equal to 144? (The answer
should be something like first, second, etc.)
d. How much will you pay for 20 acre feet of
water?
Page 8 of 9
15. Systems and Systems of Inequalities (15 pts)
−3𝑥 + 5𝑦 = −10
a. Solve the system of equations: {
𝑥+𝑦 =1
c. What are the corner points of
your regions? Be exact. (Hint:
your work in a and b is useful).
d. The objective function for this
region is z = -x + 2y. Find the
maximum value of the function
and at which point it occurs.
Show work.
b. Graph the region
determined by the
constraints. Indicate
which region it is by
shading it.
−3𝑥 + 5𝑦 ≤ −10
𝑥+𝑦 ≥1
{
𝑦≤2
𝑥 ≥ −3
16. Partial Fractions (5 pts)
Find the partial fraction decomposition of:
3𝑥+8
𝑥 2 −5𝑥−14
Page 9 of 9
17. You have finished taking the practice final. The actual final will be of similar length and
difficulty, but the problems will be different.
a. What important material did we cover, that didn’t make it into the practice exam.
I’ve started the list. Please try to add at least ten more topics. Then search for
problems in those areas and solve them.
0. Graphs of square root, cubic, absolute value functions.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
b. For every problem on the final as well as problems not, try to find 2-4 more
problems that come from the same topic, but are asked differently. For example,
Problem 2 –
 other operations on functions (function addition, subtraction, multiplication
and division);
 composition of other types of functions
 composition when the input is a number, i.e. (𝑓°𝑔)(1).
 finding inverses of other types of functions
 showing two functions are inverses
Now search for problems in the homework, on old quizzes and exams, etc. and try those to
practice these problems too.