Free printable summer review

Precalculus Summer Review

A free 40-problem Precalculus summer review PDF covering algebra, functions, trigonometry, exponentials, logarithms, sequences, vectors, analytic geometry, and Calculus readiness with a complete answer key.

40 problems9 skill areasAbout 120 minutesAnswer key included
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What this review measures

Calculus becomes much more manageable when algebra and function notation do not consume all of a student's attention. This expanded packet targets the manipulations and interpretations used immediately in limits, derivatives, and integrals, with substantial exact-value practice in functions, graphs, trigonometry, logarithms, conics, vectors, and difference quotients.

For parents and educators

This is best used as a diagnostic. Time one uninterrupted attempt, then group missed problems by section. A low score in one section is a focused summer study plan, not a verdict on calculus readiness.

Preview the complete review

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

Algebraic fluency

Factor, simplify, and solve while respecting domains.

  1. 1Factor 2x3−5x2−12x\displaystyle 2x^{3}-5x^{2}-12x completely.
  2. 2Simplify 1x+h−1xh\displaystyle \frac{\frac{1}{x+h}-\frac{1}{x}}{h}, where h≠0\displaystyle h\neq0.
  3. 3Solve 2x−1+1x+1=1\displaystyle \frac{2}{x-1}+\frac{1}{x+1}=1.
  4. 4Simplify x2−9x2−x−6⋅x−3x+3\displaystyle \frac{x^{2}-9}{x^{2}-x-6}\cdot\frac{x-3}{x+3} and state all restrictions.
  5. 5Solve ∣3x−5∣=13\displaystyle \lvert3x-5\rvert=13.

Functions

Analyze domains, compositions, inverses, and rates of change.

  1. 6Find the domain of f(x)=5−2xx+3\displaystyle f(x)=\frac{\sqrt{5-2x}}{x+3}.
  2. 7If f(x)=x2+1\displaystyle f(x)=x^{2}+1 and g(x)=x−1\displaystyle g(x)=\sqrt{x-1}, find f(g(x))\displaystyle f(g(x)) and its domain.
  3. 8Find the average rate of change of f(x)=x2−3x\displaystyle f(x)=x^{2}-3x on 1≤x≤5\displaystyle 1\leq x\leq5.
  4. 9Is f(x)=x3+2\displaystyle f(x)=x^{3}+2 one-to-one? Explain.
  5. 10Find the inverse of f(x)=3x−72\displaystyle f(x)=\frac{3x-7}{2}.

Graphs and transformations

Read key features and transformed formulas.

  1. 11Describe the transformations from y=∣x∣\displaystyle y=\lvert x\rvert to y=3∣x−2∣−5\displaystyle y=3\lvert x-2\rvert-5.
  2. 12Find the vertical and horizontal asymptotes of y=2x+1x−4\displaystyle y=\frac{2x+1}{x-4}.
  3. 13Give the center and radius of x2+y2−6x+8y=0\displaystyle x^{2}+y^{2}-6x+8y=0.
  4. 14For f(x)=x4−4x2\displaystyle f(x)=x^{4}-4x^{2}, state the zeros and describe the end behavior.

Trigonometry

Use exact values, identities, graphs, and equations.

  1. 15Find the exact value of sin⁡(5π6)cos⁡(π3)+cos⁡(5π6)sin⁡(π3)\displaystyle \sin\left(\frac{5\pi}{6}\right)\cos\left(\frac{\pi}{3}\right)+\cos\left(\frac{5\pi}{6}\right)\sin\left(\frac{\pi}{3}\right).
  2. 16Solve 2sin⁡2(x)−1=0\displaystyle 2\sin^{2}(x)-1=0 for 0≤x<2π\displaystyle 0\leq x<2\pi.
  3. 17Simplify 1−cos⁡2(x)sin⁡(x)\displaystyle \frac{1-\cos^{2}(x)}{\sin(x)}, where defined.
  4. 18For y=−3cos⁡(2x)+1\displaystyle y=-3\cos(2x)+1, state the amplitude, period, and midline.
  5. 19Find the exact value of tan⁡(7π6)\displaystyle \tan\left(\frac{7\pi}{6}\right).
  6. 20Verify the identity sec⁡2(x)−1tan⁡(x)=tan⁡(x)\displaystyle \frac{\sec^{2}(x)-1}{\tan(x)}=\tan(x) wherever both sides are defined.

Exponentials and logarithms

Solve equations and interpret continuous growth.

  1. 21Solve e2x=7\displaystyle e^{2x}=7.
  2. 22Condense 2ln⁡(x)−ln⁡(x−1)\displaystyle 2\ln(x)-\ln(x-1) into one logarithm.
  3. 23A population is modeled by P(t)=800e0.03t\displaystyle P(t)=800e^{0.03t}. Find its doubling time to the nearest tenth.
  4. 24Solve log⁡3(x+1)+log⁡3(x−3)=2\displaystyle \log_{3}(x+1)+\log_{3}(x-3)=2.

Sequences and series

Use explicit terms and finite sums.

  1. 25Find an explicit formula for 11,7,3,−1,…\displaystyle 11,7,3,-1,\ldots.
  2. 26Find the finite sum 3+6+12+⋯+384\displaystyle 3+6+12+\cdots+384.
  3. 27Does the infinite series 10+5+2.5+⋯\displaystyle 10+5+2.5+\cdots converge? If so, find its sum.
  4. 28Evaluate ∑k=112(3k−1)\displaystyle \sum_{k=1}^{12}(3k-1).

Vectors and analytic geometry

Work with magnitude, components, and conic forms.

  1. 29Find the magnitude of vector ⟨−6,8⟩\displaystyle \langle-6,8\rangle.
  2. 30Find the dot product of ⟨2,−3⟩\displaystyle \langle2,-3\rangle and ⟨5,4⟩\displaystyle \langle5,4\rangle.
  3. 31Classify 9x2+4y2=36\displaystyle 9x^{2}+4y^{2}=36 and give the vertices on its major axis.
  4. 32Write y2−8y−12x+4=0\displaystyle y^{2}-8y-12x+4=0 in standard form and identify its vertex.

Calculus readiness

Reason about change and limiting behavior without calculus rules.

  1. 33Simplify f(3+h)−f(3)h\displaystyle \frac{f(3+h)-f(3)}{h} for f(x)=x2\displaystyle f(x)=x^{2} and h≠0\displaystyle h\neq0.
  2. 34As x\displaystyle x grows without bound, what value does 3x2−1x2+4\displaystyle \frac{3x^{2}-1}{x^{2}+4} approach?
  3. 35A particle's position is s(t)=t2−4t\displaystyle s(t)=t^{2}-4t. Find its average velocity from t=1\displaystyle t=1 to t=3\displaystyle t=3.
  4. 36Simplify (x+h)3−x3h\displaystyle \frac{(x+h)^{3}-x^{3}}{h} for h≠0\displaystyle h\neq0.
  5. 37Without using a graph, describe the behavior of f(x)=1x2\displaystyle f(x)=\frac{1}{x^{2}} as x\displaystyle x approaches 0\displaystyle 0 from either side.

Challenge

Connect exact algebra to future calculus ideas.

  1. 38For x≠4\displaystyle x\neq4, simplify x2−16x−4\displaystyle \frac{x^{2}-16}{x-4}. What value would fill the hole at x=4\displaystyle x=4?
  2. 39Explain why sin⁡(x)x\displaystyle \frac{\sin(x)}{x} cannot be evaluated by direct substitution at x=0\displaystyle x=0.
  3. 40If f(x)=x+4\displaystyle f(x)=\sqrt{x+4} and g(x)=x2−4\displaystyle g(x)=x^{2}-4, compare f(g(x))\displaystyle f(g(x)) with g(f(x))\displaystyle g(f(x)) and state any domain restrictions.
View the complete answer key

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

Algebraic fluency

1. x(2x+3)(x−4)\displaystyle x(2x+3)(x-4)

2. −1x(x+h)\displaystyle -\frac{1}{x(x+h)}; x≠0\displaystyle x\neq0, x≠−h\displaystyle x\neq-h, and h≠0\displaystyle h\neq0

3. x=3+172\displaystyle x=\frac{3+\sqrt{17}}{2} or x=3−172\displaystyle x=\frac{3-\sqrt{17}}{2}

4. x−3x+2\displaystyle \frac{x-3}{x+2}; x≠3\displaystyle x\neq3, x≠−2\displaystyle x\neq-2, and x≠−3\displaystyle x\neq-3

5. x=6\displaystyle x=6 or x=−83\displaystyle x=-\frac{8}{3}

Functions

6. x≤52\displaystyle x\leq\frac{5}{2}, with x≠−3\displaystyle x\neq-3

7. f(g(x))=x\displaystyle f(g(x))=x; domain x≥1\displaystyle x\geq1

8. 3\displaystyle 3

9. Yes; it is strictly increasing, so every output corresponds to exactly one input

10. f−1(x)=2x+73\displaystyle f^{-1}(x)=\frac{2x+7}{3}

Graphs and transformations

11. Right 2\displaystyle 2, vertical stretch by 3\displaystyle 3, and down 5\displaystyle 5

12. Vertical asymptote x=4\displaystyle x=4; horizontal asymptote y=2\displaystyle y=2

13. Center (3,−4)\displaystyle (3,-4); radius 5\displaystyle 5

14. Zeros x=−2,0,2\displaystyle x=-2,0,2; f(x)\displaystyle f(x) approaches positive infinity as x\displaystyle x approaches either positive or negative infinity

Trigonometry

15. −12\displaystyle -\frac{1}{2}

16. x=π4,3π4,5π4,7π4\displaystyle x=\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}

17. sin⁡(x)\displaystyle \sin(x)

18. Amplitude 3\displaystyle 3; period π\displaystyle \pi; midline y=1\displaystyle y=1

19. 33\displaystyle \frac{\sqrt{3}}{3}

20. Use sec⁡2(x)−1=tan⁡2(x)\displaystyle \sec^{2}(x)-1=\tan^{2}(x), then tan⁡2(x)tan⁡(x)=tan⁡(x)\displaystyle \frac{\tan^{2}(x)}{\tan(x)}=\tan(x)

Exponentials and logarithms

21. x=ln⁡(7)2\displaystyle x=\frac{\ln(7)}{2}

22. ln⁡(x2x−1)\displaystyle \ln\left(\frac{x^{2}}{x-1}\right)

23. t=ln⁡(2)0.03≈23.1\displaystyle t=\frac{\ln(2)}{0.03}\approx23.1 time units

24. x=1+13\displaystyle x=1+\sqrt{13}

Sequences and series

25. an=15−4n\displaystyle a_{n}=15-4n

26. 765\displaystyle 765

27. Yes; it converges to 20\displaystyle 20

28. 222\displaystyle 222

Vectors and analytic geometry

29. 10\displaystyle 10

30. −2\displaystyle -2

31. Ellipse; (0,3)\displaystyle (0,3) and (0,−3)\displaystyle (0,-3)

32. (y−4)2=12(x+1)\displaystyle (y-4)^{2}=12(x+1); vertex (−1,4)\displaystyle (-1,4)

Calculus readiness

33. 6+h\displaystyle 6+h

34. 3\displaystyle 3

35. 0\displaystyle 0

36. 3x2+3xh+h2\displaystyle 3x^{2}+3xh+h^{2}

37. f(x)\displaystyle f(x) increases without bound from both sides

Challenge

38. x+4\displaystyle x+4; the hole would be filled by 8\displaystyle 8

39. Direct substitution gives 00\displaystyle \frac{0}{0}, an indeterminate form that requires a limiting argument

40. f(g(x))=∣x∣\displaystyle f(g(x))=\lvert x\rvert for all real x\displaystyle x; g(f(x))=x\displaystyle g(f(x))=x for x≥−4\displaystyle x\geq-4

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