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College Algebra Unit 6: Graphs & transformations

Most graphs you'll meet are transformations of a small set of parent functions. Learn the parents, then learn how each change moves them.

Covers: Intercepts, symmetry, shifts/stretches/reflections, piecewise functions

Study guide

Graphs reveal a function's behavior. Intercepts come from setting x = 0 (the y-intercept) or y = 0 (the x-intercepts). Symmetry tests: a graph is symmetric about the y-axis if f(−x) = f(x) (an even function, like x²), and about the origin if f(−x) = −f(x) (an odd function, like x³).

Transformations change a parent function's graph. Vertical shifts: f(x) + k moves up k. Horizontal shifts: f(x − h) moves right h, and f(x + h) moves left h. Reflections: −f(x) flips over the x-axis and f(−x) flips over the y-axis. Stretches: a·f(x) stretches vertically by a if a > 1 and compresses if 0 < a < 1; f(bx) compresses horizontally by 1/b. When combining transformations, apply horizontal changes inside the function and vertical changes outside, with stretches and reflections before shifts.

Know the parent graphs: linear, quadratic, cubic, square root, absolute value, reciprocal, exponential and logarithmic.

Piecewise functions use different formulas on different parts of the domain. To graph one, draw each piece only on its interval, using closed dots for included endpoints and open dots for excluded ones. To evaluate, find which interval contains the input. Absolute value |x| is a piecewise function: x for x ≥ 0 and −x for x < 0.

On a graph, f(x) > g(x) wherever the curve for f lies above the curve for g, and the x-coordinates of the intersection points solve f(x) = g(x). The distance between two points is √((x₂ − x₁)² + (y₂ − y₁)²), and the midpoint is ((x₁ + x₂)/2, (y₁ + y₂)/2). A circle with center (h, k) and radius r has equation (x − h)² + (y − k)² = r².

To model a flat fee F that covers usage up to a threshold t, plus a rate r per unit beyond it, write C(x) = F for 0 ≤ x ≤ t and C(x) = F + r(x − t) for x > t.

Key ideas

• f(x) + k shifts up k; f(x − h) shifts right h.
• −f(x) reflects over the x-axis; f(−x) reflects over the y-axis.
• a·f(x) stretches vertically by a (shrinks if 0 < a < 1).
• Even functions are symmetric about the y-axis (f(−x) = f(x)); odd functions are symmetric about the origin (f(−x) = −f(x)).
• Find x-intercepts by setting y = 0; find the y-intercept by setting x = 0.

Formulas and rules

Common mistakes

• Shifting the wrong way: f(x − 3) moves right, not left.
• Applying stretches and shifts in the wrong order.

Words to know

Effect of f(x − 4)

Shift right 4.

Effect of f(x) − 2

Shift down 2.

Effect of −f(x)

Reflect over the x-axis.

Even function

f(−x) = f(x); symmetric about the y-axis.

Odd function

f(−x) = −f(x); symmetric about the origin (e.g. x³).

X-intercept

Where y = 0.

Symmetry about the y-axis

f(−x) = f(x); an even function.

Symmetry about the origin

f(−x) = −f(x); an odd function.

Horizontal stretch/compression

f(bx) compresses by 1/b when b > 1.

Piecewise function

Different formulas on different parts of the domain.

Absolute value function

|x| = x for x ≥ 0, −x for x < 0.

Parent functions

Basic graphs like x², x³, √x, |x| and 1/x.

Sample practice questions

19 more questions like these, from level 1 to 5, are in the practice set.