Free ACT Math Exam Flashcards

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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.

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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.

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.

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.

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

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

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.

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

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.

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.

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.

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

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

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

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

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.

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

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.

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