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Topics and sample problems

Numbers

• Order of operations
  Evaluate −2² + 19.
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• Fractions
  Work out 3/7 × 7/4. Give your answer as a fraction in lowest terms.
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• Percentages
  A $200.00 jacket is 15% off, then 8% sales tax is added to the sale price. What do you pay? Round to the nearest cent.
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• Ratios and rates
  6 notebooks cost $25.20. At the same price each, what do 7 notebooks cost?
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• Exponent rules
  Simplify (x⁶)² · x¹. What is the exponent?
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• Simplifying radicals
  Simplify √180. Write it as a√b with b as small as possible.
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• Scientific notation
  Multiply (3.6 × 10⁸)(2.6 × 10⁰). Write the answer in scientific notation.

Algebra

• Linear equations
  Solve for x: 5x + 10 = 6x + 17
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• Inequalities
  Solve −6x + 11 ≥ 5.
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• Slope and lines
  A line passes through (4, 7) and (8, 10). Find its slope m and y-intercept b.
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• Systems of equations
  Solve the system:
  −3x + 4y = 41
  6x + 4y = −22
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• Factoring
  Factor 2x² − 4x − 30. Take out the common factor first.
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• Quadratic equations
  Solve: x² + x − 12 = 0
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• Functions
  f(x) = −x − 4 and g(x) = x² − 3. Find f(g(−4)).
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• Transformations
  The point (−3, 1) is on the graph of y = f(x). Where does it land on the graph of y = −2f(x + 2) + 5?
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• Exponential growth and decay
  A bacteria culture of 200 cells doubles every 5 hours. How many cells are there after 10 hours?
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• Logarithms
  Solve for x: log₅(x) = 4.
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• Sequences
  An arithmetic sequence starts 12, 10, 8, 6, … What is term number 29?
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• Absolute value
  Solve: ∣3x + 15∣ + 9 = 27
• Completing the square
  Complete the square to find the vertex of y = x² + 4x + 12.
• Polynomial division
  Divide x³ + 6x² + 8x − 3 by x − 1. Fill in the quotient and the remainder.
• Rational equations
  Solve: 7 / (x − 1) = 3 / (x − 5)
• Asymptotes
  Find the vertical and horizontal asymptotes of f(x) = (−6x + 4) / (−x − 4).
• Inverse functions
  f(x) = −3x − 1. Find f⁻¹(−13).
• Domain
  f(x) = ln(x + 0). The domain is x > what number?
• Piecewise functions
  f(x) = −x + 1 if x < −3, and 3x + 4 if x ≥ −3. Find f(−3).
• Polynomial end behavior
  f(x) = 5x². How do the ends of the graph behave?
• Complex numbers
  Multiply (−3 − i)(−2 + i). Write it as a + bi.
• Matrices
  Multiply [[4, 2], [4, 0]] × [[−2, −1], [−2, 4]].
• Word problems
  Adult tickets cost $15 and student tickets cost $5. 190 tickets were sold for $1,770. How many were adult tickets?

Geometry and trig

• Area and volume
  A cylinder has radius 9 cm and height 4 cm. Find its volume as a multiple of π.
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• Pythagorean theorem
  Find the distance between (−1, −2) and (−9, −17).
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• Circles
  A circle has radius 6 cm. Find the length of the arc cut off by a central angle of 180°. Give it as a multiple of π.
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• Right-triangle trig
  In a right triangle, angle θ = 52° and the hypotenuse is 13. Find the side opposite θ, to the nearest tenth.
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• Unit circle
  Find the exact value of sin(210°).
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• Similar triangles
  △ABC ~ △DEF. AB = 6, BC = 9 and DE = 10. Find EF.
• Law of sines and cosines
  In △ABC, a = 7, b = 12 and C = 110°. Find c to one decimal place.
• Distance and midpoint
  Find the distance between (3, 6) and (−5, 12).
• Parallel lines and angles
  Two parallel lines are cut by a transversal. Alternate interior angles measure (3x − 28)° and (2x − 4)°. Find x.
• Arc length and sectors
  A circle has radius 9 cm. Find the length of the arc cut off by a central angle of 210°. Give it as a multiple of π.
• Trig identities
  sin θ = −5/13 and θ is in quadrant IV. Find cos θ.
• Sine and cosine graphs
  Find the amplitude and the period of y = −4 sin(3x) + 2.

Statistics

• Mean, median and standard deviation
  Find the population standard deviation of: 11, 12, 4, 10, 20. Round to 2 decimal places.
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• Probability
  You roll two fair dice. What is the probability the total is 4? Give a fraction.
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• Permutations and combinations
  A club of 9 people chooses a president, vice president. No one can hold two jobs. How many ways can this be done?
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• Z-scores
  A data set has mean 45 and standard deviation 8. What value has a z-score of 0.25?
• Conditional probability
  From the table, find P(plays music | sport).

  	Plays music	No music
  Sport	35	39
  No sport	30	40

• Line of best fit
  The line of best fit is ŷ = 58 + 4.7x. A data point is (2, 62.3). Find its residual.
• Expected value
  A game costs $5 to play. You win $5 if you get a 5 or 6, otherwise nothing. Find the expected value of playing, to the nearest cent.

Calculus

• Limits
  Find the limit as x → 1 of (x² − 4x + 3) / (x − 1).
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• Derivatives
  f(x) = (−3x − 2)². Find f′(−1).
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• Integrals
  Evaluate the integral of −5x − 4 from x = 0 to x = 3.
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• Chain rule
  f(x) = (−2x − 4)². Find f′(−1).
• Product and quotient rules
  f(x) = (−x + 5) / (x − 1). Use the quotient rule to find f′(−3).
• Related rates
  A 17 ft ladder leans on a wall. Its base slides away at 4 ft/s. How fast is the top sliding down when the base is 8 ft from the wall?

Test prep

• SAT math mix
  Solve for x: 6x + 13 = 7x + 18
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• ACT math mix
  Solve for x: 4x + 4 = 9x + 19
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Science

• Motion graphs
  A velocity–time graph is a straight line from 3 m/s at t = 0 to 13 m/s at t = 5 s. What is the acceleration?
• Vectors
  Find the magnitude of the vector ⟨−5, 12⟩.
• Kinematics
  A car starts at 0 m/s and accelerates at 2.5 m/s² for 4 s. Find its final velocity.