CLEP Precalculus cheat sheet

Quadratic formula: for ax² + bx + c = 0 with a ≠ 0, x = (−b ± √(b² − 4ac)) / (2a). The discriminant b² − 4ac determines the number of real roots: positive gives two, zero gives one, negative gives none.

Factoring identities: a² − b² = (a − b)(a + b); a³ ± b³ = (a ± b)(a² ∓ ab + b²); perfect-square trinomials a² ± 2ab + b² = (a ± b)².

Rational expressions: simplify by factoring numerator and denominator and canceling common factors; the domain excludes values that make any original denominator zero.

Absolute value inequalities: |x| < k (k > 0) is equivalent to −k < x < k; |x| > k is equivalent to x < −k or x > k. For |ax + b| ≥ k, solve ax + b ≥ k or ax + b ≤ −k.

Polynomial division: remainder theorem says the remainder of p(x) divided by x − c is p(c); factor theorem says x − c is a factor exactly when p(c) = 0.

Exponential and logarithmic equations: if b^x = b^y then x = y; log_b M = log_b N implies M = N (M, N > 0). To solve a^x = c, take logarithms: x = log c / log a.

Rational inequalities: move all terms to one side, combine into a single fraction, find zeros and undefined points, and test signs on the resulting intervals; never multiply by an expression of unknown sign.

Systems of equations: substitution isolates one variable; elimination combines equations to cancel a variable. A linear–quadratic system can have 0, 1, or 2 real solutions.

Inequality rules: multiplying or dividing by a negative number reverses the inequality sign; adding or subtracting the same quantity preserves it.

A function assigns exactly one output to each input in its domain. The vertical line test determines whether a graph represents a function; the horizontal line test determines whether it is one-to-one.

Domain restrictions: exclude values making a denominator zero and require radicands of even roots to be nonnegative; for logarithms require the argument positive.

Composition: (f ∘ g)(x) = f(g(x)); the domain of f ∘ g consists of x in the domain of g with g(x) in the domain of f. Composition is not generally commutative.

Inverse functions: f has an inverse function only if it is one-to-one. f⁻¹ undoes f: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x; the graph of f⁻¹ is the reflection of f across y = x; domain and range swap.

Transformations: y = a·f(b(x − h)) + k stretches vertically by |a|, horizontally by 1/|b|, shifts right h and up k; negative a reflects across the x-axis, negative b across the y-axis.

Even and odd functions: f is even if f(−x) = f(x) (symmetric about the y-axis); f is odd if f(−x) = −f(x) (symmetric about the origin).

Average rate of change of f on [a, b] is (f(b) − f(a)) / (b − a), the slope of the secant line through the endpoints.

Combining functions: (f + g)(x) = f(x) + g(x), (fg)(x) = f(x)g(x), (f/g)(x) = f(x)/g(x) with g(x) ≠ 0; domains are intersections of the individual domains with any extra restrictions.

Linear functions: slope-intercept form y = mx + b; point-slope form y − y₁ = m(x − x₁); slope m = (y₂ − y₁)/(x₂ − x₁).

Quadratic functions: standard form y = ax² + bx + c; vertex form y = a(x − h)² + k with vertex (h, k); vertex x-coordinate h = −b/(2a).

Polynomial end behavior: for a leading term aₙxⁿ, as x → ±∞ the graph resembles aₙxⁿ; even degree gives same-direction tails, odd degree gives opposite-direction tails.

Rational functions: vertical asymptotes occur where the denominator is zero but the numerator is not (after canceling common factors); horizontal asymptotes depend on degrees: numerator less than denominator gives y = 0; equal degrees give ratio of leading coefficients; numerator greater gives none (unless a slant asymptote).

Exponential functions: y = a·b^x with b > 1 models growth, 0 < b < 1 models decay; y-intercept is (0, a); horizontal asymptote y = 0.

Logarithmic functions: y = log_b x is the inverse of y = b^x; domain x > 0, range all reals, vertical asymptote x = 0, x-intercept (1, 0).

From a table, a constant first difference indicates a linear function; a constant ratio of successive outputs indicates an exponential function.

Zeros of a function are the x-intercepts of its graph, the solutions of f(x) = 0, and the x-values where the graph crosses or touches the x-axis.

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²). Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).

Circle: center (h, k), radius r gives (x − h)² + (y − k)² = r²; complete the square to convert general form to standard form.

Parabola: y = a(x − h)² + k opens upward if a > 0, downward if a < 0; x = a(y − k)² + h opens right if a > 0, left if a < 0.

Ellipse: (x − h)²/a² + (y − k)²/b² = 1 has center (h, k); major axis length 2a along the larger denominator, minor axis length 2b.

Hyperbola: (x − h)²/a² − (y − k)²/b² = 1 opens horizontally with asymptotes y − k = ±(b/a)(x − h); reversed signs open vertically with asymptotes y − k = ±(a/b)(x − h).

Slopes: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1 (negative reciprocals), except vertical and horizontal lines.

Line equations: standard form Ax + By = C; slope-intercept y = mx + b; vertical line through (a, b) is x = a; horizontal line is y = b.

Right-triangle definitions: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent; csc, sec, cot are reciprocals.

Unit circle: for angle θ in standard position, (cos θ, sin θ) is the point on the unit circle; tan θ = sin θ / cos θ.

Pythagorean identity: sin²θ + cos²θ = 1; derived identities 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ.

Radian and degree conversion: π radians = 180°; arc length s = rθ for θ in radians; area of a sector A = ½r²θ.

Law of Sines: a/sin A = b/sin B = c/sin C; Law of Cosines: c² = a² + b² − 2ab cos C.

Sine and cosine graphs: y = A sin(B(x − C)) + D has amplitude |A|, period 2π/|B|, phase shift C, vertical shift D; tangent has period π/|B| and no amplitude.

Inverse trig functions: sin⁻¹ x has range [−π/2, π/2]; cos⁻¹ x has range [0, π]; tan⁻¹ x has range (−π/2, π/2).

Solving trigonometric equations: find all solutions in one period, then add multiples of the period; check for extraneous solutions when squaring.

Linear model: constant rate of change; y = mx + b where m is the rate and b the initial value.

Exponential model: constant percent change; y = a·b^x where a is the initial amount and b = 1 + r (growth) or 1 − r (decay) for rate r per period.

Compound interest: A = P(1 + r/n)^(nt), where P is principal, r annual rate, n compounding periods per year, t years; continuous compounding A = Pe^(rt).

Logistic growth: y = L / (1 + Ae^(−kt)) with carrying capacity L; initial value L/(1 + A); growth is fast in the middle and levels off.

Inverse variation: y = k/x; direct variation: y = kx; joint variation combines multiple direct and inverse factors.

Half-life: amount remaining after t is A = A₀(1/2)^(t/h) where h is half-life; doubling time from exponential growth satisfies b^T = 2.

Model selection: use finite differences to detect polynomial degree (constant nth differences for degree n) and ratios of successive outputs to detect exponential behavior.

Interpreting parameters: in a model, the coefficient usually represents an initial quantity or scale, while the base or rate represents growth/decay per unit time.