MB Pre-Calculus 40S flashcards

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

Transformation mapping notation

For y = a·f(b(x − h)) + k, a point (x, y) on the base graph maps to (x/b + h, a·y + k). This single rule combines stretches, reflections, and translations in one step.

Even versus odd function

A function is even when f(−x) = f(x), giving symmetry about the y-axis. It is odd when f(−x) = −f(x), giving symmetry about the origin.

Inverse of a function

The inverse relation swaps x and y, so its graph reflects the original across y = x. The inverse is a function only if the original passes the horizontal line test.

Reciprocal graph features

For y = 1/f(x), zeros of f become vertical asymptotes, and points where f(x) = 1 or −1 stay fixed. Large values of f produce values near zero.

Absolute value transformation

The graph of y = |f(x)| keeps every part of f above the x-axis and reflects each part below the x-axis upward, creating a graph that is never negative.

Extraneous roots in radical equations

Squaring both sides can create solutions that do not satisfy the original equation. Every candidate root must be substituted back, and failures are discarded.

Rational function asymptote rules

Compare degrees of numerator and denominator. Smaller numerator degree gives y = 0, equal degrees give the ratio of leading coefficients, and a larger numerator gives no horizontal asymptote.

Hole versus vertical asymptote

A factor shared by numerator and denominator cancels and creates a hole at that x-value. A remaining denominator factor that is not cancelled gives a vertical asymptote.

Radical function domain and endpoint

For y = a√(b(x − h)) + k, the radicand must be non-negative. The graph begins or ends at the point (h, k), and the range lies on one side of k.

Radian measure conversions

Multiply degrees by π/180 to obtain radians, and multiply radians by 180/π to obtain degrees. A full rotation equals 2π radians or 360 degrees.

Arc length and sector area

With central angle θ in radians and radius r, arc length is s = rθ and sector area is A = ½r²θ. Both formulas require radian measure, not degrees.

Sinusoid parameters

In y = a sin b(x − c) + d, amplitude is |a|, period is 2π/|b|, phase shift is c, and d is the vertical displacement of the axis.

CAST rule quadrants

All six trigonometric ratios are positive in quadrant I. Sine is positive in II, tangent in III, and cosine in IV; the others are negative there.

Law of Cosines

a² = b² + c² − 2bc·cos A relates three sides to one angle. It is used for SAS and SSS triangles, where the sine law alone is insufficient.

Ambiguous case of sine law

When given two sides and a non-included angle, zero, one, or two triangles may exist. Check whether the supplementary of the computed angle keeps the angles summing below 180 degrees.

Logarithm product and quotient laws

The log of a product equals the sum of the logs, log(MN) = log M + log N. The log of a quotient equals the difference, log(M/N) = log M − log N.

Change of base formula

log_b M = log M / log b allows evaluation in any base. Choose base 10 or base e for calculator use when the given base is not available.

Exponential growth and decay

In y = a·b^x, the value grows when b > 1 and decays when 0 < b < 1. The y-intercept is a and the horizontal asymptote is y = 0.

Logarithmic function properties

The graph of y = log_b x has domain x > 0, range all real numbers, a vertical asymptote at x = 0, and an x-intercept at 1.

Compound interest formula

A = P(1 + r/n)^(nt) gives the amount after t years when interest rate r is compounded n times yearly. Continuous growth uses A = Pe^(rt).

Solving logarithmic equations

Condense the equation to a single logarithm, rewrite in exponential form, and solve. Any candidate that makes an argument zero or negative is rejected.

Fundamental counting principle

If one choice can be made in m ways and a second in n ways, the two together can be made in m × n ways. Extend by multiplying all stages.

Permutation formula

nPr = n!/(n − r)! counts ordered arrangements of r items chosen from n distinct items. Order matters, so different orders count as different arrangements.

Combination formula

nCr = n!/(r!(n − r)!) counts unordered selections of r items from n. Order does not matter, so each group is counted only once.

Permutations with identical items

When n objects include a identical items of one kind and b of another, the number of distinct arrangements is n!/(a!·b!·…). Dividing removes duplicate orderings.

General term of binomial expansion

In (x + y)^n, the term containing y^k is nCk · x^(n−k) · y^k. The index k runs from 0 to n, giving n + 1 terms.

Circle equation standard form

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

Parabola focal length

For (x − h)² = 4p(y − k), the vertex is (h, k) and the focus lies p units from the vertex along the axis of symmetry.

Ellipse axis relationship

For an ellipse, a is the semi-major axis, b the semi-minor axis, and c the focal distance, related by c² = a² − b². Foci lie on the major axis.

Hyperbola asymptotes

For (x − h)²/a² − (y − k)²/b² = 1, the asymptotes are y − k = ±(b/a)(x − h). The foci satisfy c² = a² + b².

Arithmetic series sum

An arithmetic series with first term t₁ and common difference d has sum S_n = (n/2)(2t₁ + (n − 1)d), equivalent to (n/2)(t₁ + t_n).

Infinite geometric series convergence

An infinite geometric series sums to t₁/(1 − r) only when |r| < 1. If |r| ≥ 1, the terms do not shrink and the series diverges.