Free printable summer review

Algebra 2 Summer Review

A free 40-problem Algebra 2 summer review PDF covering functions, polynomials, rational and radical expressions, complex numbers, exponentials, logarithms, systems, sequences, probability, and trigonometry with a complete answer key.

40 problems8 skill areasAbout 110 minutesAnswer key included
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What this review measures

Precalculus expects students to see functions as objects that can be transformed, composed, inverted, and analyzed. This expanded review prioritizes those connections while refreshing polynomial structure, exact radical and complex-number work, logarithmic reasoning, nonlinear systems, sequences, probability, and the trigonometric skills that make advanced function work possible.

For parents and educators

Students should write domain restrictions before canceling rational expressions and check every proposed solution to radical or rational equations in the original equation.

Preview the complete review

Every question below appears in the printable packet. Students should show their work and simplify answers unless directed otherwise.

Functions and transformations

Analyze, compose, invert, and transform functions.

  1. 1For f(x)=2x−3\displaystyle f(x)=2x-3 and g(x)=x2+1\displaystyle g(x)=x^{2}+1, find g(f(2))\displaystyle g(f(2)).
  2. 2Find the inverse of f(x)=x−53\displaystyle f(x)=\frac{x-5}{3}.
  3. 3Describe the transformations from y=x2\displaystyle y=x^{2} to y=−2(x+4)2+7\displaystyle y=-2(x+4)^{2}+7.
  4. 4For f(x)=x2−4\displaystyle f(x)=x^{2}-4 and g(x)=1x+2\displaystyle g(x)=\frac{1}{x+2}, find f(g(0))\displaystyle f(g(0)).
  5. 5Simplify the difference quotient f(x+h)−f(x)h\displaystyle \frac{f(x+h)-f(x)}{h} for f(x)=x2+3x\displaystyle f(x)=x^{2}+3x and h≠0\displaystyle h\neq0.

Polynomial functions

Factor, solve, and use polynomial structure.

  1. 6Factor x3−4x2−x+4\displaystyle x^{3}-4x^{2}-x+4 completely.
  2. 7Divide 2x3+3x2−11x−6\displaystyle 2x^{3}+3x^{2}-11x-6 by x−2\displaystyle x-2.
  3. 8List all zeros of x3−3x2+4x−12\displaystyle x^{3}-3x^{2}+4x-12.
  4. 9State the end behavior of f(x)=−3x4+2x\displaystyle f(x)=-3x^{4}+2x.
  5. 10Use the Remainder Theorem to find the remainder when 3x4−2x3+x−5\displaystyle 3x^{4}-2x^{3}+x-5 is divided by x+2\displaystyle x+2.

Rational and radical expressions

Track domains and reject extraneous solutions.

  1. 11Simplify x2−9x2+x−6\displaystyle \frac{x^{2}-9}{x^{2}+x-6} and state all restrictions.
  2. 12Solve 3x−1=2x+2\displaystyle \frac{3}{x-1}=\frac{2}{x+2}.
  3. 13Solve 2x+3=x\displaystyle \sqrt{2x+3}=x and check for extraneous solutions.
  4. 14Simplify (3+2i)(4−i)\displaystyle (3+2i)(4-i).
  5. 15Rationalize the denominator of 52−3\displaystyle \frac{5}{2-\sqrt{3}}.
  6. 16Solve x+6=x−2\displaystyle \sqrt{x+6}=x-2 and check the proposed solutions.

Exponential and logarithmic functions

Convert forms and solve growth equations.

  1. 17Solve 5x+1=125\displaystyle 5^{x+1}=125.
  2. 18Expand log⁡2(8x3y)\displaystyle \log_{2}\left(\frac{8x^{3}}{y}\right).
  3. 19Solve log⁡3(x−1)+log⁡3(x+1)=2\displaystyle \log_{3}(x-1)+\log_{3}(x+1)=2.
  4. 20An investment of $1,200 grows by 5%\displaystyle 5\% annually. Find its value after 4\displaystyle 4 years to the nearest cent.
  5. 21Solve 2e3x=11\displaystyle 2e^{3x}=11.

Equations and systems

Solve nonlinear equations and mixed systems.

  1. 22Solve x4−13x2+36=0\displaystyle x^{4}-13x^{2}+36=0.
  2. 23Solve the system y=x2\displaystyle y=x^{2} and y=2x+3\displaystyle y=2x+3.
  3. 24Solve ∣2x+1∣>7\displaystyle \lvert2x+1\rvert>7.
  4. 25Solve the system x+y=5\displaystyle x+y=5 and x2+y2=13\displaystyle x^{2}+y^{2}=13.
  5. 26Solve x2+6x+13=0\displaystyle x^{2}+6x+13=0 over the complex numbers.

Sequences and probability

Use explicit formulas and counting principles.

  1. 27Find the sum of the first 20\displaystyle 20 terms of 5,8,11,…\displaystyle 5,8,11,\ldots.
  2. 28A geometric sequence has a1=6\displaystyle a_{1}=6 and r=12\displaystyle r=\frac{1}{2}. Find a6\displaystyle a_{6}.
  3. 29How many 4\displaystyle 4-letter arrangements can be made from 7\displaystyle 7 distinct letters without repetition?
  4. 30Find the coefficient of x2\displaystyle x^{2} in (x+3)4\displaystyle (x+3)^{4}.
  5. 31Determine whether 12−4+43−49+⋯\displaystyle 12-4+\frac{4}{3}-\frac{4}{9}+\cdots converges. If it does, find the sum.

Trigonometry preview

Recall unit-circle values and solve basic triangles.

  1. 32Give the exact values of sin⁡(π3)\displaystyle \sin\left(\frac{\pi}{3}\right) and cos⁡(π3)\displaystyle \cos\left(\frac{\pi}{3}\right).
  2. 33Find all x\displaystyle x with 0≤x<2π\displaystyle 0\leq x<2\pi such that cos⁡(x)=−12\displaystyle \cos(x)=-\frac{1}{2}.
  3. 34A triangle has sides 7\displaystyle 7 and 10\displaystyle 10 with an included angle of 60\displaystyle 60 degrees. Find the exact third side.
  4. 35Simplify 1−cos⁡2(x)sin⁡(x)\displaystyle \frac{1-\cos^{2}(x)}{\sin(x)}, where defined.
  5. 36Solve 2sin⁡(x)=3\displaystyle 2\sin(x)=\sqrt{3} for 0≤x<2π\displaystyle 0\leq x<2\pi.

Challenge

Connect representations and restrictions.

  1. 37If f(x)=x2−4\displaystyle f(x)=x^{2}-4 is restricted to x≥0\displaystyle x\geq0, find f−1(x)\displaystyle f^{-1}(x).
  2. 38Why can x=−1\displaystyle x=-1 not solve log⁡2(x+2)+log⁡2(x)=1\displaystyle \log_{2}(x+2)+\log_{2}(x)=1?
  3. 39Solve 2x−3−1x+3=9x2−9\displaystyle \frac{2}{x-3}-\frac{1}{x+3}=\frac{9}{x^{2}-9} and state the excluded values.
  4. 40For f(x)=2x\displaystyle f(x)=2^{x} and g(x)=log⁡2(x)\displaystyle g(x)=\log_{2}(x), simplify f(g(x))\displaystyle f(g(x)) and g(f(x))\displaystyle g(f(x)), including domain restrictions.
View the complete answer key

Open-ended explanations may use different wording while showing the same reasoning.

Functions and transformations

1. 2\displaystyle 2

2. f−1(x)=3x+5\displaystyle f^{-1}(x)=3x+5

3. Left 4\displaystyle 4, vertical stretch by 2\displaystyle 2, reflection across the x\displaystyle x-axis, and up 7\displaystyle 7

4. −154\displaystyle -\frac{15}{4}

5. 2x+h+3\displaystyle 2x+h+3

Polynomial functions

6. (x−4)(x−1)(x+1)\displaystyle (x-4)(x-1)(x+1)

7. 2x2+7x+3\displaystyle 2x^{2}+7x+3

8. 3,2i,−2i\displaystyle 3,2i,-2i

9. f(x)\displaystyle f(x) approaches negative infinity as x\displaystyle x approaches either positive or negative infinity

10. 57\displaystyle 57

Rational and radical expressions

11. x−3x−2\displaystyle \frac{x-3}{x-2}; x≠−3\displaystyle x\neq-3 and x≠2\displaystyle x\neq2

12. x=−8\displaystyle x=-8

13. x=3\displaystyle x=3

14. 14+5i\displaystyle 14+5i

15. 10+53\displaystyle 10+5\sqrt{3}

16. x=5+332\displaystyle x=\frac{5+\sqrt{33}}{2}

Exponential and logarithmic functions

17. x=2\displaystyle x=2

18. 3+3log⁡2(x)−log⁡2(y)\displaystyle 3+3\log_{2}(x)-\log_{2}(y)

19. x=10\displaystyle x=\sqrt{10}

20. $1,458.61\displaystyle 1,458.61

21. x=ln⁡(112)3\displaystyle x=\frac{\ln\left(\frac{11}{2}\right)}{3}

Equations and systems

22. x=−3,−2,2,3\displaystyle x=-3,-2,2,3

23. (−1,1)\displaystyle (-1,1) and (3,9)\displaystyle (3,9)

24. x<−4\displaystyle x<-4 or x>3\displaystyle x>3

25. (2,3)\displaystyle (2,3) and (3,2)\displaystyle (3,2)

26. x=−3±2i\displaystyle x=-3\pm2i

Sequences and probability

27. 670\displaystyle 670

28. 316\displaystyle \frac{3}{16}

29. 840\displaystyle 840

30. 54\displaystyle 54

31. Yes; the common ratio is −13\displaystyle -\frac{1}{3}, and the sum is 9\displaystyle 9

Trigonometry preview

32. 32\displaystyle \frac{\sqrt{3}}{2} and 12\displaystyle \frac{1}{2}

33. x=2π3\displaystyle x=\frac{2\pi}{3} and x=4π3\displaystyle x=\frac{4\pi}{3}

34. 79\displaystyle \sqrt{79}

35. sin⁡(x)\displaystyle \sin(x)

36. x=π3\displaystyle x=\frac{\pi}{3} and x=2π3\displaystyle x=\frac{2\pi}{3}

Challenge

37. f−1(x)=x+4\displaystyle f^{-1}(x)=\sqrt{x+4}

38. log⁡2(x)\displaystyle \log_{2}(x) is undefined when x≤0\displaystyle x\leq0

39. x=0\displaystyle x=0; excluded values x≠−3,3\displaystyle x\neq-3,3

40. f(g(x))=x\displaystyle f(g(x))=x for x>0\displaystyle x>0; g(f(x))=x\displaystyle g(f(x))=x for every real x\displaystyle x

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