Reciprocal, Exponential, Trigonometric Graphs and Transformations

Algebra • Graphs

Flashcards

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What point does y = k^x always pass through?

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The point (0, 1).

Key concepts

What you'll likely be quizzed about

Reciprocal function y = 1/x

The reciprocal function y = 1/x has domain x ≠ 0 and range y ≠ 0. Vertical asymptote occurs at x = 0 and horizontal asymptote at y = 0, because division by values near zero produces very large magnitude outputs and values of x with large magnitude produce outputs near zero. The graph is a rectangular hyperbola with one branch in quadrant I (x > 0, y > 0) and one branch in quadrant III (x < 0, y < 0), showing odd symmetry about the origin.

Exponential functions y = k^x (k > 0)

Exponential functions always pass through the point (0, 1) because any positive base to the power zero equals 1. The horizontal asymptote is y = 0 because k^x approaches zero as x → −∞ when 0 < k < 1, and as x → −∞ for k > 1 the value still tends to 0. If k > 1 the graph shows exponential growth: small increases in x produce multiplicative increases in y. If 0 < k < 1 the graph shows exponential decay: increases in x produce multiplicative decreases in y. The function is always positive for all real x.

Trigonometric functions y = sin x, y = cos x, y = tan x (degrees)

The sine and cosine functions have amplitude 1 and period 360°, with sin x and cos x values in the range [−1, 1]. Cosine is maximum at x = 0 (cos 0 = 1) and sine is zero at x = 0 (sin 0 = 0); both repeat every 360°. The tangent function has period 180°, is unbounded and has vertical asymptotes at angles where cos x = 0 (x = 90° + 180°·n) because tan x = sin x / cos x. Sine and cosine graphs are smooth waves; tangent graphs have repeating S-shaped curves between asymptotes.

Translations: shifting a graph

A vertical translation y = f(x) + a moves the graph up by a when a > 0 and down by |a| when a < 0 because every y-value increases by a. A horizontal translation y = f(x − a) shifts the graph right by a when a > 0 and left by |a| when a < 0 because each x input requires adjustment to produce the same output. Translations preserve shape, amplitude and period but change coordinates of intercepts and asymptotes.

Reflections: flipping a graph

A reflection in the x-axis y = −f(x) multiplies all y-values by −1, so peaks become troughs and the graph inverts vertically. A reflection in the y-axis y = f(−x) reverses left-right orientation, so points at x become points at −x; even functions (like cos x) remain unchanged while odd functions (like sin x and 1/x) change sign. Reflections preserve shape and size but reverse orientation.

Key notes

Important points to keep in mind

y = 1/x has vertical asymptote x = 0 and horizontal asymptote y = 0; branches in QI and QIII.

Exponential graphs always pass through (0,1) and stay positive for all real x.

k > 1 gives growth; 0 < k < 1 gives decay.

sin and cos have amplitude 1 and period 360°; tan has period 180° and vertical asymptotes.

Vertical translation y = f(x) + a moves graph up or down by a without changing shape.

Horizontal translation y = f(x − a) moves graph right by a; use inputs shifted by a.

Reflection y = −f(x) flips vertically; reflection y = f(−x) flips horizontally.

Apply transformations in order: inside changes (x) first, then outside changes (y).

Asymptotes move with horizontal shifts; horizontal asymptotes remain unless limits change.

Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.