MATH 106 WVC Compute the Values Using Properties of Logarithms Questions

Final ExamMath 106, West Valley College, Fall 2021
You may use one 3” by 5” notecard and a non-graphing calculator for the test. Carefully read all the
instructions and clearly show all your work. Box your final answers. Good Luck!!!☺
1. (7 points each) Simplify completely:
a.
b.
1
𝑥 2 +8𝑥+15
1
𝑥−1
+
÷
2
𝑥−5
3
𝑥+5
6𝑥 −5 𝑦
2
c. ( 2 −4 )
2𝑥 𝑦
d. 7√9𝑥 9 + 𝑥 4 √25𝑥
2. (7 points) Use the properties of logarithms to expand as much as possible:
log 3 (
√(𝑥+1)
27
)
3. (15 points) A ball is thrown straight up from a height of 3 feet. The ball’s height in feet above the
ground, t seconds after it is thrown, can be modeled by the function
ℎ(𝑡) = −𝑡 2 + 2𝑡 + 3
a. Write and solve a quadractic equation to find when the ball will hit the ground. Round your
answer to one decimal.
b. When will the ball reach the maximum height?
c. What will be the maximum height of the ball?
4. (7 points each) Multiply and Simplify:
a.
(2 − 5√3)(2 + 5√3)
b. (7 − 4𝑖)2
5. (10 points) Find the vertex and intercepts of the given quadratic function. Then use it to sketch
the graph of the quadratic function. Use the
graph to identify the range of the quadratic
function.
𝑔(𝑥) = (𝑥 − 1)2 − 9
a. Vertex:
b. 𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑠):
c. 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡(𝑠):
d. Domain:
e. Range:
6. (6 points) Find the equation of the line passing through the point (−6, 5) and parallel to the line
1
whose equation is 𝑦 = 𝑥 + 5.
3
7. (6 points) Solve the linear inequality:
3(𝑥 + 5) − 5 < 6𝑥 − 3(𝑥 − 3) 8. (10 points) The formula 𝐴 = 36.1𝑒 0.0126𝑡 models the population of California, A, in millions, t years after 2005. a. Use the model to estimate the population in the California in 2012. b. When will the population reach 40 million? 9. (7 points) Find the restrictions and solve: 3𝑥 4 + =3 𝑥+1 𝑥−2 10. (6 points) Solve the given system of equations (write your final answer as a co-ordinate): 2𝑥 + 3𝑦 = 10 3𝑥 + 𝑦 = 1 11. (8 points each) Solve: a. 𝑙𝑜𝑔𝑥 = 4 b. 72𝑥−1 = 49 12. (6 points each) Find the indicated function values for the function 𝑓(𝑥) = 4𝑥 − 2. a. 𝑓(−5) b. 𝑓(𝑎 + ℎ) 13. (10 points) Find the 𝑥 − 𝒂𝒏𝒅 𝑦 −intercepts: 3𝑥 − 4𝑦 = 36. 14. (7 points) Find the domain and write it in the interval notation: 𝑔(𝑥) = √2𝑥 − 10 15. (8 points) Using the slope and the 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 to graph: 4 𝑦 =− 𝑥−1 5 16. (8 points) Solve using the Quadratic Formula and leave your answer in the exact form: 2𝑥 2 − 𝑥 = 5 17. (7 points) Let 𝑓(𝑥) = −2𝑥 − 5 and 𝑔(𝑥) = 𝑥 2 + 3𝑥. Find (𝑓 ∘ 𝑔)(𝑥) 18. (7 points each) Factor completely: a. 4𝑥 4 𝑦 5 + 28𝑥 3 𝑦 3 + 16𝑥 2 𝑦 9 b. 𝑥 3 − 15𝑥 2 − 𝑥 + 15 c. 4𝑥 2 − 8𝑥 − 5 19. (10 points) Answer the questions after analyzing the graph given below. You may assume that the graph continues at the right end. a. Determine if 𝑦 is a function of 𝑥. You must give reason. b. Find the domain of the function. Write your answer in interval notation. c. Find the range of the function. Write your answer in interval notation. YAY, you are DONE!!!! Wait, did you check your work? Happy Holidays!!!