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Summer brush-up · Math

Algebra II check: ready for Trigonometry or Statistics?

For students who finished Algebra II (usually grades 10–11) or are deciding whether to start Trigonometry or Statistics.

14 items drawn from the Algebra II final exam. About 30–40 minutes. Paper and pencil, no calculator unless an item says so. Work every item first, then check each answer and mark it.

Print test with answer key

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1. Let f(x) = x^2 + 1 and g(x) = \sqrt{x - 3}. Find (f \circ g)(x) in simplest form and give its domain.

Answer: f(g(x)) = (\sqrt{x - 3})^2 + 1 = x - 2, domain [3, \infty) (the input must first be allowed in g)
Lesson 1.4 Combining and Composing Functions

2. Write a quadratic function f(x) = x^2 + bx + c whose zeros are 2 + i and 2 - i.

Answer: Sum of roots 4, product (2 + i)(2 - i) = 5, so f(x) = x^2 - 4x + 5
Lesson 3.4 Complex Solutions and the Discriminant

3. Solve x^3 - x^2 - 6x < 0. Answer in interval notation.

Answer: x(x - 3)(x + 2) < 0; a sign chart on the zeros -2, 0, 3 gives (-\infty, -2) \cup (0, 3)
Lesson 4.7 Graphing Polynomial Functions

4. Henry builds wooden birdhouses. His tools and setup cost $600, and each birdhouse costs $8 in materials, so his average cost per birdhouse is \bar{C}(x) = \dfrac{600 + 8x}{x}. (a) How many birdhouses must he build to bring the average cost down to $10? (b) What value does the average cost approach as x grows large?

Answer: (a) 600 + 8x = 10x, so x = 300 birdhouses; (b) the horizontal asymptote is y = 8: the average cost approaches $8 per birdhouse
Lesson 5.3 Rational Functions and Asymptotes

5. Challenge. A boat travels 24 miles upstream and 24 miles back in a total of 5 hours. Its speed in still water is 10 miles per hour. Find the speed of the current.

Answer: \dfrac{24}{10 - c} + \dfrac{24}{10 + c} = 5, so 480 = 5(100 - c^2), c^2 = 4, and c = 2 mph (check: 3 + 2 = 5 hours ✓)
Lesson 5.5 Solving Rational Equations

6. By hand, simplify (8x^6y^{-3})^{2/3}, assuming x > 0 and y > 0. Write the answer with positive exponents.

Answer: 8^{2/3}x^{4}y^{-2} = \dfrac{4x^4}{y^2}
Lesson 6.2 Properties of Rational Exponents and Radicals

7. The period of a pendulum is T = 2\pi\sqrt{\dfrac{L}{32}} seconds, where L is its length in feet. A rope swing in the barn is 8 feet long. Find its period to the nearest hundredth of a second.

Answer: T = 2\pi\sqrt{\dfrac{8}{32}} = 2\pi \cdot \dfrac{1}{2} = \pi \approx 3.14 seconds
Lesson 6.4 Radical Functions and Their Graphs

8. Ruth deposits $5,000 at 4% annual interest for 10 years. Find the balance to the nearest cent if the interest is (a) compounded continuously, (b) compounded quarterly.

Answer: (a) 5000e^{0.4} \approx \$7459.12; (b) 5000(1.01)^{40} \approx \$7444.32
Lesson 7.2 Compound Interest and the Number e

9. Evaluate by hand. (a) \log_3 81 (b) \log_5 \dfrac{1}{125} (c) \log_4 8

Answer: (a) 4, since 3^4 = 81; (b) -3, since 5^{-3} = \dfrac{1}{125}; (c) \dfrac{3}{2}, since 4^{3/2} = 8
Lesson 8.1 Logarithms

10. Samuel stacks firewood logs in rows. The bottom row has 20 logs, each row above has one fewer, and the top row has 6 logs. (a) How many rows are there? (b) How many logs are in the stack?

Answer: (a) 6 = 20 + (n - 1)(-1), so n = 15 rows; (b) S_{15} = \dfrac{15(20 + 6)}{2} = 195 logs
Lesson 9.2 Arithmetic Sequences and Series

11. A church is forming a committee for a service project: 3 adults chosen from 8 adult volunteers and 2 teenagers chosen from 6 teen volunteers. How many different committees are possible?

Answer: C(8, 3) \cdot C(6, 2) = 56 \times 15 = 840 committees
Lesson 10.2 Combinations

12. For the parabola (x - 2)^2 = 8(y + 1), find the vertex, the focus and the directrix.

Answer: 4p = 8, so p = 2; vertex (2, -1), focus (2, 1), directrix y = -3
Lesson 11.1 Parabolas: Focus and Directrix

13. A farm stand sold 10 dozen eggs and 4 jars of honey on Friday, and 6 dozen eggs and 8 jars of honey on Saturday. Eggs are $4 a dozen and honey is $9 a jar. Find Q P for Q = \begin{bmatrix} 10 & 4 \\ 6 & 8 \end{bmatrix} and P = \begin{bmatrix} 4 \\ 9 \end{bmatrix}, and say what its entries mean.

Answer: QP = \begin{bmatrix} 10(4) + 4(9) \\ 6(4) + 8(9) \end{bmatrix} = \begin{bmatrix} 76 \\ 96 \end{bmatrix}: the stand took in $76 on Friday and $96 on Saturday
Lesson 12.2 Matrix Multiplication

14. Round to the nearest tenth. (a) A 24-foot ladder leans against a barn and makes a 65^\circ angle with the ground. How high up the wall does it reach? (b) A shed roof rises 4 feet over a horizontal run of 12 feet. What angle does it make with the horizontal?

Answer: (a) 24\sin 65^\circ \approx 21.8 feet; (b) \tan\theta = \dfrac{4}{12}, so \theta \approx 18.4^\circ
Lesson 13.1 Right-Triangle Trigonometry: A Review

4-week summer review path

Four short days a week, 30–45 minutes: the 16 Algebra II lessons that appear most often on its exams and reviews, in course order. Do the lessons you missed on the test first.

Week 1

  1. Day 1: Lesson 1.4 Combining and Composing Functions · worksheet A · key
  2. Day 2: Lesson 2.4 Systems of Three Equations in Three Variables · worksheet A · key
  3. Day 3: Lesson 3.4 Complex Solutions and the Discriminant · worksheet A · key
  4. Day 4: Lesson 4.7 Graphing Polynomial Functions · worksheet A · key
  5. Day 5: retake the items on this test from these lessons; correct every mistake.

Week 2

  1. Day 1: Lesson 5.3 Rational Functions and Asymptotes · worksheet A · key
  2. Day 2: Lesson 5.5 Solving Rational Equations · worksheet A · key
  3. Day 3: Lesson 6.2 Properties of Rational Exponents and Radicals · worksheet A · key
  4. Day 4: Lesson 6.4 Radical Functions and Their Graphs · worksheet A · key
  5. Day 5: retake the items on this test from these lessons; correct every mistake.

Week 3

  1. Day 1: Lesson 6.5 Solving Radical Equations · worksheet A · key
  2. Day 2: Lesson 7.2 Compound Interest and the Number e · worksheet A · key
  3. Day 3: Lesson 7.3 Exponential Models · worksheet A · key
  4. Day 4: Lesson 7.4 Logistic Functions · worksheet A · key
  5. Day 5: retake the items on this test from these lessons; correct every mistake.

Week 4

  1. Day 1: Lesson 8.1 Logarithms · worksheet A · key
  2. Day 2: Lesson 8.4 Solving Exponential Equations · worksheet A · key
  3. Day 3: Lesson 8.5 Solving Logarithmic Equations · worksheet A · key
  4. Day 4: Lesson 8.6 Logarithmic Scales and Models · worksheet A · key
  5. Day 5: retake the items on this test from these lessons; correct every mistake.