Algebra 2 Name______
Spring Semester Exam Review
1. What is the inverse function of ?
2. What is the domain of ?
3. Write the equation of with the following transformations:
a) a vertical shift of 2 units up
b) 4 units to the left
c) a vertical dilation by a factor of 3
d) a reflection across the x-axis
4. Graph . Make sure to label at least 3 points that are easy to find when you make the appropriate transformations.
5. Pilots use the function to approximate the distance D, in kilometers to the horizon from an altitude A in meters.
a) What is the approximate distance to the horizon observed by a pilot flying at an altitude of 11,000 meters?
b) How will the approximate distance appear to change if the pilot descends by 4,000 meters?
6. Solve the following radical functions:
a) b)
c) d)
State whether each function represents exponential growth or decay. Then state the growth or decay factor as a percent.
7. 8.
Graph each of the following exponential functions (help yourself to graph paper!). State the transformations applied to the parent function . Then state the y-intercept, asymptote, domain, and range.
9. 10.
11. The population P(t) of a bacteria colony t hours after being put in a dish can be approximated by the function .
a) Is the population increasing or decreasing? By what factor?
b) Predict the bacteria population after 3, 8, and 12 hours.
12. A new car is purchased for $21,500. Its value each year is about 85% of the value of the preceding year.
a) Write an equation to model this situation (identify your variables). Then sketch the graph.
b) Find the value of the car after 2 years, 5 years, and 10 years.
c) When will the car’s value be zero? Explain your answer.
Complete the table.
Intital Amount / Interest Rate / Compounding / Time (years) / Function / Amount13. $8000 / 3.5% / Annually / 1
14. $7000 / 4.125% / Monthly / 2
15. $6000 / 5.25% / Continuously / 8
16. The table gives the approximate number of grams of element X at different times, in years.
Time / 1 / 10 / 15 / 25 / 40Grams / 96 / 67 / 55 / 37 / 20
a) The equation that models this situation is . How many grams were there at time 0?
b) How many grams will be there in 100 years?
17. Rewrite each expression using logarithmic notation.
a) 33 = 27 b) 20 = 1
18. Rewrite each expression using exponential notation.
a) log 2 32 = 5 b) ln e = 1
19. Sketch the graph of f(x) = log 3 x.
20. Sketch the graph of g(x) = log 0.5 x.
21. Expand each logarithm.
a) ln 12 b) log 25
22. Condense each logarithm.
a) log 2 + log 5 b) log 3 + log x – log 4
23. Solve log 2 (3x) = 5.
24. Solve 4 log x = 2.
25. Find the inverse of y = 2x + 1.
26. Find the inverse of y = 4x + 5.
27. State whether the equation represents direct variation, inverse variation, or neither.
a) b) c) d)
28. Find the constant of variation, a, and write the equation of variation
a) y varies inversely as x and when
b) y varies directly as x and when
29. The weight, w, of a certain material varies directly with the surface area, a, of that material. For this material, 8 square feet weighs 0.5 pounds.
a) Find the constant of variation, k, and write the equation of variation.
b) How much will 10 square feet of this material weigh?
30. The time, t, it takes a runner to cover a specified distance varies inversely as the runner’s speed, s. If Larry Long runs at an average of 6 miles per hour, he will finish the race in 1.4 hours.
a) Find the constant of variation, k, and write the equation of variation.
b) How long will is take Sam Short to finish the race if he runs at an average of 5 miles per hour?
31. With relation to the parent function, , state the transformations that have been made to each function:
a) , b)
32. Write the inverse variation function that has:
a) vertical stretch of 5, shifted right 3 and down 4
b) vertical stretch of 2, shifted up 3 and flipped.
33. For each of the following functions, state the equations of the horizontal and vertical asymptote and state the domain and range:
a) b) c) d)
34. Simplify the rational expression: a) b)
35. Subtract the polynomials (3x3 + 5x2 – 6x) – (x3 + 3x2 – 8)
36. Add the polynomials
37. Multiply the polynomials
38. Multiply the polynomials (3x – 5)(2x2 + 2x – 7)
39. Write the following in radical notation
a. b. c.
40. Write the following in rational exponent notation
a. b. c.
41. Simplify:
a. b. c. d.
42. Solve by factoring:
a. b. c. d.
43. Use synthetic division: ( – 4x + 7) ÷ (x – 2)
44. Factor 27 – 1 completely.