ACT Math flashcards
ACT Math · Build recall with a focused review session.
32 cards. Read the term and the definition here. The deck at the top of this page is optional.
- Order of operations
- Evaluate in this sequence: parentheses and grouping symbols, then exponents and roots, then multiplication and division from left to right, then addition and subtraction from left to right.
- Negative and zero exponents
- For nonzero x, x^0 = 1, and x^(−n) = 1/x^n. A negative exponent signals a reciprocal, not a negative value.
- Fractional exponent conversion
- x^(m/n) equals the n-th root of x raised to the m power. So x^(1/2) = √x and x^(2/3) = the cube root of x squared.
- Percent change formula
- Percent change = (new value − old value) ÷ old value × 100. A positive result is an increase and a negative result is a decrease.
- Powers of i
- Since i = √(−1) and i² = −1, powers of i repeat in a four-step cycle: i, −1, −i, 1. Divide the exponent by 4 and use the remainder.
- Scientific notation rules
- Write numbers as a × 10^n with 1 ≤ |a| < 10. To multiply, multiply the coefficients and add exponents; to divide, divide coefficients and subtract exponents.
- Slope formula
- m = (y₂ − y₁)/(x₂ − x₁) gives the slope between two points. Horizontal lines have slope 0; vertical lines have undefined slope.
- Quadratic formula
- x = (−b ± √(b² − 4ac))/(2a) gives the solutions of ax² + bx + c = 0 when a ≠ 0, covering both real and complex roots.
- Discriminant test
- For ax² + bx + c = 0, the value b² − 4ac tells the root type: positive means two real roots, zero means one repeated root, negative means no real roots.
- Parabola vertex location
- In y = ax² + bx + c, the vertex sits at x = −b/(2a). Substitute that x to find the maximum or minimum y-value.
- Perpendicular slope relationship
- Two lines are perpendicular exactly when their slopes are negative reciprocals, so m₁ · m₂ = −1. Parallel lines have identical slopes.
- Difference of squares
- a² − b² factors into (a − b)(a + b). Recognizing this pattern lets you factor quickly and cancel factors in rational expressions.
- Function composition
- (f ∘ g)(x) = f(g(x)) means apply g first, then feed that output into f. Order matters, so f(g(x)) usually differs from g(f(x)).
- Finding an inverse function
- Swap x and y in y = f(x), then solve for y to get f⁻¹. Graphically, an inverse reflects the original function across the line y = x.
- Horizontal shift rule
- Replacing x with (x − h) moves a graph right by h units, while (x + h) moves it left by h units. The sign inside acts opposite to intuition.
- Vertical stretch and reflection
- Multiplying outside the function by a > 1 stretches the graph vertically; by 0 < a < 1 compresses it; by −1 reflects it across the x-axis.
- Exponential growth and decay
- In f(x) = a·b^x, the value a is the starting amount. If b > 1 the function grows; if 0 < b < 1 it decays toward zero.
- Average rate of change
- On the interval from a to b, the average rate of change is (f(b) − f(a))/(b − a), the slope of the secant line joining those two points.
- Even and odd functions
- A function is even when f(−x) = f(x), giving y-axis symmetry. It is odd when f(−x) = −f(x), giving symmetry about the origin.
- Triangle angle sum
- The three interior angles of any triangle total 180°. An exterior angle equals the sum of the two interior angles that are not adjacent to it.
- Similar triangle proportion
- Similar triangles have equal corresponding angles and proportional corresponding sides. Set corresponding sides as a ratio and cross-multiply to find a missing length.
- Special right triangle sides
- A 45-45-90 triangle has sides x, x, x√2. A 30-60-90 triangle has sides x, x√3, 2x opposite the 30°, 60°, and 90° angles.
- Arc length and sector area
- Arc length = (central angle/360) × 2πr, and sector area = (central angle/360) × πr². Both are fractions of the whole circle.
- Circle equation standard form
- (x − h)² + (y − k)² = r² describes a circle with center (h, k) and radius r. Completing the square converts expanded equations to this form.
- Volume formulas
- Prisms and cylinders use V = Bh; cones and pyramids use V = (1/3)Bh; a sphere uses V = (4/3)πr³, where B is base area.
- Distance and midpoint
- Distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²); the midpoint is the average of the x-coordinates and of the y-coordinates.
- Mean median and mode
- The mean is the sum divided by the count, the median is the middle value of ordered data, and the mode is the most frequent value.
- Independent event multiplication
- For independent events, P(A and B) = P(A) × P(B). For mutually exclusive events, P(A or B) = P(A) + P(B).
- Complement probability rule
- P(not A) = 1 − P(A). An event and its complement always have probabilities that add to exactly 1.
- Expected value calculation
- Expected value is the sum of each outcome multiplied by its probability, giving the long-run average result of a repeated random event.
- Permutations versus combinations
- Permutations count ordered arrangements (nPr), while combinations count unordered selections. The relationship is nCr = nPr ÷ r!.
- Interquartile range definition
- IQR = Q3 − Q1, the range covering the middle 50% of data. On a box plot, values beyond 1.5 × IQR from the quartiles are outliers.