Algebra 2 Combined Review
Name: ________________________
1. Find the point-slope form of the equation of the line passing through the points (–6, –4) and
(2, –5).
2. Write the equation for the translation of y = |x|.
3. Write the equation that is the translation of y = |x| left 2 units and up 2 units.
Solve the system.
4. x − y = −4
2x − 2y = −8
6.
4x + 2y = 20
7x − 3y = 9
5.
x + y = −6
x − 4y = −1
7.
−x − 3y = 4
x + 3y = 7
8. A system of two linear inequalities ____ has a solution.
a. always b. sometimes c. never
9. Given the system of constraints, name all vertices. Then find the maximum value of the given
objective function.
x≥ 0
y≥ 0
6x − 2y ≤ 12
Maximum for C = 4x − 3y
4y ≤ 4x + 8
10. Dalco Manufacturing estimates that its weekly profit, P, in hundreds of dollars, can be
approximated by the formula P = −3x2 + 6x + 10, where x is the number of units produced
per week, in thousands.
a. How many units should the company produce per week to earn the maximum profit?
b. Find the maximum weekly profit.
11. Graph y = −2(x − 2)2 – 4.
12. Use vertex form to write the equation of the parabola.
13. Identify the vertex and the y-intercept of the graph of the function y = −3(x + 2)2 + 5.
Factor the expression.
14. 4x2 + 20x + 25
15. 4x2 − 25
16. Solve by factoring. 4x2 − 30x − 100 = 0
Solve the equation by finding square roots.
17. 3x2 = 21
Simplify the expression.
18. (−2 + i) + (−2 − 2i)
19. (1 + 3i) − (6 + 2i)
20. (−8i)(4i)
21. (−5 − 5i)(−2 + 6i)
Solve the equation.
22. 36x2 + 9 = 0
23. √𝑥 + 9 − 10 = −7
24. x2 + 4x − 3 = 0
25. x2 + 6x + 16 = 0
26. −2x2 + x + 8 = 0
27. Find the zeros of y = x(x + 4)(x + 3). Then graph the equation.
28. Write a polynomial function in standard form with zeros at 5, –3, and 1.
29. Find the zeros of f(x) = (x + 5)2(x − 4)4 and state the multiplicity.
30. The numbers of cookies in a shipment of bags are normally distributed, with a mean of 64
and a standard deviation of 4. What percent of bags of cookies will contain between 60 and 68
cookies?
31. A bag contains 6 red marbles, 6 white marbles, and 4 blue marbles. Find P(red or blue).
Simplify the radical expression. Use absolute value symbols if needed.
4
32. √625𝑥 20 𝑦 8
Multiply and simplify if possible.
33. √15 ∙ √5
4
4
34. √11 ∙ √−11
3
35. Simplify √135𝑎10 𝑏 9. Assume that all variables are positive.
Simplify.
36. −√6 + 6√9 + 4√6
Multiply.
37. (7 − √3)(8 + √3)
Solve. Check for extraneous solutions.
38. 5𝑥 = √20 − 5𝑥
39. Let f(x) = 3x + 5 and g(x) = 4x + 7. Find f(x) + g(x).
40. Let f(x) = 3x + 2 and g(x) = −7x − 6. Find f ∙g and its domain.
41. Let f(x) = −4x − 5 and g(x) = −2x − 6. Find (f ° g)(−3).
Write the equation in logarithmic form.
42. 72 = 49
Evaluate the logarithm.
43. log5
1
44. log7343
25
Write the expression as a single logarithm.
45. 4 logbv + 3 logbx
46. log280 – log210
Expand the logarithmic expression.
47. log 7
𝑛
2
48. Solve ln(3x − 1) = 1. Round to the nearest thousandth.
49. Solve ln 2 + ln x = 5.
Simplify the rational expression. State any restrictions on the variable.
𝑎2 +3𝑎−28
50.
𝑎+7
Multiply or divide. State any restrictions on the variables.
51.
𝑔2
𝑔+2
∙
𝑔2 −3𝑔−10
𝑔2 −6𝑔
Add or subtract. Simplify if possible.
52.
4
𝑔−9
+
1
𝑔2 −81
Solve the equation. Check the solution.
53.
−2
𝑥−4
=
5
54.
𝑥−3
𝑦+7
𝑦−8
=
𝑦+8
𝑦−3
55. Graph the system of constraints. Then find the values of x and y that maximize P = 40x + 0y.
x≥ 0
y≥ 0
−2x + 2y ≤ 4
x≤ 3
56. Graph y = 2x2 − 7.
57. In a baseball game, an outfielder throws a ball to the second baseman. The path of the ball is
1
375 2
705
modeled by the equation = −
(𝑥 − 2 ) + 16 , where y is the height of the ball in feet
900
after the ball has traveled x feet horizontally. The second baseman catches the ball at the same
height as the height at which the outfielder released it.
a. What was the maximum height of the ball along its path? Answer to the nearest foot.
b. How far was the second baseman from the outfielder at the time he caught the ball?
c. How high above the ground was the ball when it left the hand of the outfielder?
58. Use the graph of y = (x − 3)2 + 5.
a. If you translate the parabola to the right 2 units and down 7 units, what is the equation of
the new parabola in vertex form?
b. If you translate the original parabola to the left 2 units and up 7 units, what is the equation
of the new parabola in vertex form?
c. How could you translate the new parabola in part (a) to get the new parabola in part (b)?
𝑥−2
59. Let f(x) = 4 and g(x) = 2x2 + 4.
a. Find f(g(x)).
b. Find g(f(x)).
60. A model for the height of a toy rocket shot from a platform is y = −16x2 + 145x + 7, where x
is the time in seconds and y is the height in feet.
a. Graph the function.
b. Find the zeros of the function.
c. What do the zeros represent? Are they realistic?
d. About how high does the rocket fly before hitting the ground? Explain.
61. Sketch a normal curve with a mean of 50 and a standard deviation of 2. Label the horizontal
axis at one, two, and three standard deviations from the mean.