CLEP Precalculus flashcards

33 cards. Read the term and the definition here. The deck at the top of this page is optional.

Quadratic formula

For ax² + bx + c = 0 with a ≠ 0, the solutions are x = (−b ± √(b² − 4ac)) / (2a). The expression under the radical is the discriminant.

Discriminant

The quantity b² − 4ac for a quadratic ax² + bx + c. If positive, two distinct real roots; if zero, one real root; if negative, no real roots.

Remainder theorem

When a polynomial p(x) is divided by x − c, the remainder equals p(c). Consequently, x − c is a factor exactly when p(c) = 0.

Absolute value inequality rule

For k > 0, |x| < k means −k < x < k, while |x| > k means x < −k or x > k. Replace x with any expression and solve the resulting compound inequality.

Rational inequality method

Move all terms to one side to form a single fraction, find zeros of numerator and denominator, mark them on a number line, and test the sign of the fraction on each interval.

Composition of functions

(f ∘ g)(x) = f(g(x)). The domain consists of all x in the domain of g such that g(x) lies in the domain of f. Composition is not commutative in general.

Inverse function conditions

A function has an inverse function only if it is one-to-one. The inverse satisfies f(f⁻¹(x)) = x and f⁻¹(f(x)) = x, and its graph reflects f across the line y = x.

Even and odd functions

A function is even if f(−x) = f(x) for all x in its domain, giving symmetry about the y-axis. It is odd if f(−x) = −f(x), giving symmetry about the origin.

Average rate of change

For a function f on the interval [a, b], the average rate of change is (f(b) − f(a)) / (b − a). It equals the slope of the secant line through the two endpoints.

Vertical line test

A graph represents a function if and only if no vertical line intersects it more than once. This ensures each input value has at most one output value.

Horizontal line test

A function is one-to-one if and only if no horizontal line intersects its graph more than once. Only one-to-one functions have inverse functions defined on their full range.

Vertex of a parabola

For y = ax² + bx + c, the vertex has x-coordinate −b/(2a). In vertex form y = a(x − h)² + k, the vertex is (h, k). The parabola opens upward if a > 0.

Polynomial end behavior

The end behavior of a polynomial is determined by its leading term aₙxⁿ. Even-degree polynomials have tails pointing the same direction; odd-degree polynomials have tails pointing opposite directions.

Horizontal asymptote rules

For a rational function, compare degrees of numerator and denominator. If numerator degree is less, y = 0. If equal, y equals the ratio of leading coefficients. If greater, no horizontal asymptote.

Exponential function form

y = a·b^x with b > 0 and b ≠ 1. If b > 1 the function models growth; if 0 < b < 1 it models decay. The y-intercept is (0, a) and y = 0 is a horizontal asymptote.

Logarithmic function properties

y = log_b x is the inverse of y = b^x. Its domain is x > 0, its range is all real numbers, its graph has vertical asymptote x = 0 and x-intercept (1, 0).

Linear function from a table

A table of values represents a linear function when the first differences between consecutive outputs are constant. That constant equals the slope of the line.

Distance formula

The distance between points (x₁, y₁) and (x₂, y₂) in the coordinate plane is √((x₂ − x₁)² + (y₂ − y₁)²). It follows from the Pythagorean theorem.

Midpoint formula

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

Circle equation

A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r². Completing the square converts a general quadratic equation into this standard form.

Perpendicular line slopes

Two nonvertical lines are perpendicular exactly when their slopes are negative reciprocals, so their product is −1. A horizontal line is perpendicular to a vertical line.

Ellipse standard equation

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

Hyperbola asymptotes

For (x − h)²/a² − (y − k)²/b² = 1, the asymptotes are y − k = ±(b/a)(x − h). If the y² term is positive, the asymptotes use slope ±(a/b).

Unit circle definitions

For an angle θ in standard position, the point where its terminal side meets the unit circle is (cos θ, sin θ). Thus tan θ = sin θ / cos θ wherever cos θ ≠ 0.

Pythagorean identity

For every angle θ, sin²θ + cos²θ = 1. Dividing by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = csc²θ.

Law of Cosines

In a triangle with sides a, b, c opposite angles A, B, C, c² = a² + b² − 2ab cos C. It applies to any triangle and reduces to the Pythagorean theorem when C = 90°.

Law of Sines

In any triangle, a/sin A = b/sin B = c/sin C, where each side is opposite the angle with the same letter. It is useful when two angles and a side are known.

Amplitude and period

For y = A sin(B(x − C)) + D, the amplitude is |A| and the period is 2π/|B|. The graph is shifted horizontally by C and vertically by D.

Inverse sine range

The inverse sine function sin⁻¹ x has domain [−1, 1] and range [−π/2, π/2]. Its output is the angle in that interval whose sine is x.

Compound interest formula

A = P(1 + r/n)^(nt) gives the amount after t years when principal P earns annual rate r compounded n times per year. Continuous compounding uses A = Pe^(rt).

Exponential growth model

y = a·b^t where a is the initial amount and b = 1 + r for growth rate r per time period. If 0 < b < 1, the model represents decay instead.

Half-life model

If a quantity has half-life h, the amount remaining after time t is A = A₀(1/2)^(t/h). After each interval of length h, the amount is halved.

Logistic growth model

y = L / (1 + Ae^(−kt)) models growth with carrying capacity L. The initial value is L/(1 + A), and the graph levels off at y = L as t increases.