ACT Math Exam Flashcards

Q: How many questions are on ACT Math?

The enhanced ACT Math section has 45 multiple-choice questions in 50 minutes. Most questions count toward the section score; a small number may be embedded field-test items.

Q: What formulas should I memorize for ACT Math?

Prioritize slope, distance, midpoint, quadratic formula, exponent rules, area and volume formulas, circle equations, special right triangle ratios, basic trigonometric ratios, probability rules, and percent change.

Q: Is a calculator allowed on ACT Math?

Yes. ACT Math permits an approved calculator for the full section. You still need fast mental setup because many questions test which formula or rule applies before calculation begins.

Q: What topics are most important on ACT Math?

ACT Math emphasizes Preparing for Higher Math: algebra, functions, geometry, statistics and probability, and number and quantity, plus integrating essential skills such as rates, percentages, and proportional reasoning.

Q: How should I use ACT Math flashcards?

Use flashcards to recall the rule first, then immediately apply it to a short problem. ACT Math rewards recognizing the structure of a question quickly under time pressure.

Q: What ACT Math score is considered college ready?

ACT's college-readiness benchmark for math is commonly reported as 22 on the 1-36 scale. Colleges and programs can set their own expectations.

Question 1

Linear equations: what is the safest solving order?

Accepted Answer

Undo operations in reverse order: clear parentheses, combine like terms, move variable terms to one side, move constants to the other side, then divide by the coefficient.

Question 2

When do you flip an inequality sign?

Accepted Answer

Flip the inequality only when multiplying or dividing both sides by a negative number. Adding, subtracting, or multiplying by a positive number does not change the direction.

Question 3

Slope formula from two points

Accepted Answer

Slope is rise over run: m = (y2 - y1)/(x2 - x1). Keep the point order consistent in the numerator and denominator.

Question 4

Slope-intercept form

Accepted Answer

y = mx + b, where m is the slope and b is the y-intercept. The graph crosses the y-axis at (0, b).

Question 5

Systems of equations: when is elimination efficient?

Accepted Answer

Use elimination when adding or subtracting the equations can cancel one variable. If coefficients are not opposites, multiply one or both equations first.

Question 6

Factoring x^2 + bx + c

Accepted Answer

Find two numbers that multiply to c and add to b. Those numbers become the constants in (x + m)(x + n).

Question 7

Quadratic formula

Accepted Answer

For ax^2 + bx + c = 0, x = (-b +/- sqrt(b^2 - 4ac))/(2a). Use it when factoring is slow or not obvious.

Question 8

What does the discriminant tell you?

Accepted Answer

The discriminant is b^2 - 4ac. Positive means two real solutions, zero means one repeated real solution, and negative means no real solutions.

Question 9

Difference of squares pattern

Accepted Answer

a^2 - b^2 factors as (a - b)(a + b). The pattern works only for subtraction, not addition.

Question 10

Radical equations: what final step is easy to forget?

Accepted Answer

Check proposed solutions in the original equation. Squaring both sides can create extraneous solutions.

Question 11

Function notation f(a)

Accepted Answer

f(a) means the output of the function when the input is a. Substitute a for every x in the rule before simplifying.

Question 12

Domain restrictions to check first

Accepted Answer

Denominators cannot be zero, even roots cannot have negative radicands in real-number questions, and logarithm inputs must be positive.

Question 13

Function composition order

Accepted Answer

(f o g)(x) means f(g(x)): do g first, then put that result into f. Work from the inside out.

Question 14

How do inverse functions switch inputs and outputs?

Accepted Answer

An inverse reverses a function's ordered pairs. To find a formula, replace f(x) with y, swap x and y, then solve for y.

Question 15

Vertical and horizontal shifts

Accepted Answer

f(x) + k shifts the graph up k units. f(x - h) shifts it right h units, because the input must increase by h to produce the old output.

Question 16

Exponential growth form

Accepted Answer

A common model is y = a(1 + r)^t, where a is the initial amount, r is the growth rate per period, and t is the number of periods.

Question 17

What does a logarithm mean?

Accepted Answer

log_b(a) = c means b^c = a. A logarithm is the exponent needed to turn the base into the given value.

Question 18

Piecewise functions: what decides the rule?

Accepted Answer

The input value determines which piece applies. Check the interval or condition first, then use only that rule.

Question 19

Distance formula

Accepted Answer

The distance between (x1, y1) and (x2, y2) is sqrt((x2 - x1)^2 + (y2 - y1)^2). It comes from the Pythagorean theorem.

Question 20

Midpoint formula

Accepted Answer

The midpoint of (x1, y1) and (x2, y2) is ((x1 + x2)/2, (y1 + y2)/2). Average the x-coordinates and y-coordinates separately.

Question 21

Parallel and perpendicular slopes

Accepted Answer

Parallel lines have equal slopes. Perpendicular nonvertical lines have slopes whose product is -1, so each slope is the negative reciprocal of the other.

Question 22

Pythagorean theorem

Accepted Answer

In a right triangle, a^2 + b^2 = c^2, where c is the hypotenuse. The hypotenuse is always opposite the right angle.

Question 23

45-45-90 triangle ratio

Accepted Answer

The sides are x, x, x sqrt(2). The hypotenuse is a leg multiplied by sqrt(2).

Question 24

30-60-90 triangle ratio

Accepted Answer

The sides are x, x sqrt(3), 2x. The shortest side is opposite 30 degrees; the hypotenuse is twice the shortest side.

Question 25

Similar figures and scale factor

Accepted Answer

Corresponding lengths scale by k, areas scale by k^2, and volumes scale by k^3. Do not use the same scale for every measurement type.

Question 26

Circle circumference and area

Accepted Answer

Circumference is 2 pi r or pi d. Area is pi r^2. Circumference measures boundary length; area measures the region inside.

Question 27

Circle equation in standard form

Accepted Answer

(x - h)^2 + (y - k)^2 = r^2 has center (h, k) and radius r. Watch the signs inside the parentheses.

Question 28

Cylinder volume

Accepted Answer

Volume of a cylinder is pi r^2 h. Think base area times height: the circular base has area pi r^2.

Question 29

SOH-CAH-TOA

Accepted Answer

sin = opposite/hypotenuse, cos = adjacent/hypotenuse, and tan = opposite/adjacent for a chosen acute angle in a right triangle.

Question 30

Radians to degrees

Accepted Answer

Multiply radians by 180/pi to convert to degrees. Multiply degrees by pi/180 to convert to radians.

Question 31

Unit circle coordinates and trig values

Accepted Answer

On the unit circle, a point has coordinates (cos theta, sin theta). The x-coordinate gives cosine and the y-coordinate gives sine.

Question 32

Pythagorean trig identity

Accepted Answer

sin^2 x + cos^2 x = 1. It is useful for converting between sine and cosine when one value is known.

Question 33

When is Law of Sines useful?

Accepted Answer

Use a/sin A = b/sin B = c/sin C when you know an angle-side opposite pair and need another side or angle in a non-right triangle.

Question 34

Mean vs. median

Accepted Answer

The mean is the arithmetic average and is affected by outliers. The median is the middle value and is usually more resistant to extreme values.

Question 35

Weighted average

Accepted Answer

Multiply each value by its weight, add the products, then divide by the total weight. This prevents small groups and large groups from counting equally.

Question 36

What does standard deviation measure?

Accepted Answer

Standard deviation measures spread around the mean. A larger standard deviation means the data values are more dispersed.

Question 37

Complement probability

Accepted Answer

The probability that an event does not happen is 1 - P(event). Complements are useful for questions asking for at least one success.

Question 38

Independent event probability

Accepted Answer

For independent events, multiply probabilities: P(A and B) = P(A)P(B). Independence means one event does not change the probability of the other.

Question 39

Permutation vs. combination

Accepted Answer

Use permutations when order matters and combinations when order does not matter. Arrangements are permutations; groups or committees are combinations.

Question 40

Conditional probability

Accepted Answer

P(A given B) means restrict the sample space to cases where B already happened, then find the fraction of those cases where A also happens.

Question 41

Negative exponent rule

Accepted Answer

a^-n = 1/a^n for nonzero a. A negative exponent moves the factor across the fraction bar; it does not make the value negative.

Question 42

Rational exponent rule

Accepted Answer

a^(m/n) means the nth root of a raised to the m power. For example, a^(1/2) is the square root of a.

Question 43

Multiplying powers with the same base

Accepted Answer

Keep the base and add the exponents: a^m * a^n = a^(m+n). This rule applies only when the bases match.

Question 44

Complex number fact to remember

Accepted Answer

i^2 = -1. Use this to simplify powers of i and products involving imaginary numbers.

Question 45

Arithmetic vs. geometric sequences

Accepted Answer

Arithmetic sequences add a constant difference. Geometric sequences multiply by a constant ratio.

Question 46

Matrix multiplication dimensions

Accepted Answer

An m by n matrix can multiply an n by p matrix, and the result is m by p. The inner dimensions must match.

Question 47

Absolute value meaning

Accepted Answer

Absolute value is distance from zero, so it is never negative. Equations like |x - a| = b usually split into two cases when b is positive.

Question 48

Percent change formula

Accepted Answer

Percent change = (new - original)/original * 100%. Positive means increase; negative means decrease.

Question 49

Unit rate setup

Accepted Answer

A unit rate puts the quantity per 1 unit, such as miles per hour or dollars per item. Divide the total amount by the number of units.

Question 50

Direct proportion test

Accepted Answer

Two quantities are directly proportional when their ratio stays constant. The model has the form y = kx, where k is the constant of proportionality.

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