MATH 3 San Diego State University Algebra Final Exam Practice

Math 3—College Algebra—Final Exam—Summer 2021Instructor: Su Lan Wong
Section # 53218
Name: ____________________________________
Q1. (6 points) For (𝑥 + 3)2 + (𝑦 − 4)2 = 25
a) Find the center (ℎ, 𝑘) and radius 𝑟 of the circle.
b) Graph the circle
𝑥−4
Q2. (6 points) Solve: 2𝑥+1 ≤ 5, write the solution set in interval notation.
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Math 3—College Algebra—Final Exam—Summer 2021
Section # 53218
Instructor: Su Lan Wong
Name: ____________________________________
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Q3. (6 points) Let 𝑓(𝑥) = 4𝑥 − 𝑥 𝑎𝑛𝑑 𝑔(𝑥) = 2𝑥 − 5. Find and simplify each of the
following expressions.
b)
a) (𝑓𝑜𝑔)(𝑥)
Q4. (6 points) Let 𝑓(𝑥) =
𝑔(𝑥+ℎ)−𝑔(𝑥)
ℎ
2𝑥−1
𝑥+5
, find 𝑓 −1 (𝑥). Give the domain and range of both 𝑓 𝑎𝑛𝑑 𝑓 −1.
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Math 3—College Algebra—Final Exam—Summer 2021
Section # 53218
Instructor: Su Lan Wong
Name: ____________________________________
Q5. (6 points) An equation of a hyperbola is given. (a) Find the vertices, foci and asymptotes of
the hyperbola. (b) sketch a graph of the hyperbola.
𝑦2 𝑥2
−
=1
9 16
Q6. (5 points) Evaluate the sum if possible.
1 1
1
1+ +
+
+⋯
6 36 216
Q7. (5 points) Expand the binomial by using the binomial theorem.
(2𝑥 + 𝑦 2 )5
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Math 3—College Algebra—Final Exam—Summer 2021
Section # 53218
Instructor: Su Lan Wong
Name: ____________________________________
Q8. (5 points) The initial size of a culture of bacteria is 8600. After 1 hour the bacteria count is
10,000.
a) Find a function that models the number of bacteria 𝑛(𝑡) after t hours.
b) After how many hours will the number of bacteria doubles?
Q9. (5 points) Solve the logarithmic equation for 𝑥.
𝑙𝑜𝑔9 (𝑥 − 5) + 𝑙𝑜𝑔9 (𝑥 + 3) = 1
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Math 3—College Algebra—Final Exam—Summer 2021
Section # 53218
Instructor: Su Lan Wong
Name: ____________________________________
Q10. (6 points) Find the partial fraction decomposition
6𝑥 − 7
(𝑥 − 2)(𝑥 + 3)
Q11. (6 points) Solve the system by using addition method.
{
3𝑥 2 − 𝑦 2 = −4
𝑥 2 + 2𝑦 2 = 36
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Math 3—College Algebra—Final Exam—Summer 2021
Instructor: Su Lan Wong
Section # 53218
Name: ____________________________________
Q12. (8 points) Given 𝑃(𝑥) = 𝑥 4 + 𝑥 3 + 7𝑥 2 + 9𝑥 − 18
a) List all possible rational zeros (without testing to see whether they actually are zeros)
b) Determine the possible number of positive and negative real zeros using Descartes’ Rule
of Signs.
c) Is (𝑥 + 2) a factor of 𝑃(𝑥)?
d) Find all complex zeros of 𝑃(𝑥)
e) Sketch the graph of P.
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Math 3—College Algebra—Final Exam—Summer 2021
Instructor: Su Lan Wong
Q13. (8 points) Given
a)
b)
c)
d)
e)
𝑟(𝑥 ) =
Section # 53218
Name: ____________________________________
𝑥+2
𝑥 2 +2𝑥−3
,
Find the domain
Find 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡𝑠 𝑎𝑛𝑑 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡
Find vertical asymptotes and horizontal asymptotes
Sketch the graph of 𝑟(𝑥)
Solve the inequality 𝑟(𝑥) ≥ 0
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Math 3—College Algebra—Final Exam—Summer 2021
Instructor: Su Lan Wong
Name: ____________________________________
Q14. (8 points) Given 𝑓(𝑥) = 𝑒 𝑥 + 3
a)
b)
c)
d)
e)
Section # 53218
Sketch the graph of the function
State its domain and range in interval notation
Find 𝑓 −1 (𝑥), and state its domain and range.
Graph 𝑓 −1 (𝑥) on the same graph as part (a).
State the asymptote for 𝑓(𝑥) 𝑎𝑛𝑑 𝑓 −1 (𝑥)
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Math 3—College Algebra—Final Exam—Summer 2021
Instructor: Su Lan Wong
Section # 53218
Name: ____________________________________
Q15. ( 6 points) Graph the solution set. If there is no solution, indicate that the solution set is the
empty set.
{
𝑦 ≥ 𝑥2
𝑥+𝑦 ≥6
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Math 3—College Algebra—Final Exam—Summer 2021
Section # 53218
Instructor: Su Lan Wong
Name: ____________________________________
Q16. (8 points) Determine whether the equation represents an ellipse, a parabola, or a hyperbola.
If the graph is an ellipse, find the center, foci, and vertices. If it is a parabola, find the vertex,
focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then
sketch the graph of the equation.
9𝑥 2 − 54𝑥 + 𝑦 2 + 2𝑦 = −46
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