1.12 Transformations of Functions

Shifting and Reflecting

For a function f(x), adding/subtracting outside shifts vertically: f(x)+k moves up k, f(x)−k moves down k. Inside the parentheses shifts horizontally (opposite direction): f(x−h) moves right h, f(x+h) moves left h. Negating outside, −f(x), reflects over the x-axis; negating inside, f(−x), reflects over the y-axis.

Stretching and Compressing

Multiplying outside by a: a·f(x) stretches vertically if |a|>1, compresses if 0<|a|<1. Multiplying the input by b: f(bx) compresses horizontally if |b|>1, stretches if 0<|b|<1 — horizontal changes act inversely to their coefficient's size.

Combining Transformations

General form a·f(b(x−h))+k combines all transformations. Apply in order: horizontal shift/stretch inside parentheses first (affecting x-values), then vertical stretch and shift outside. Identifying a, b, h, k from a graph or equation lets you predict key features like intercepts, asymptotes, and extrema of the transformed function.

Key terms

f(x) + k

Vertical shift: up k units if k > 0, down |k| units if k < 0.

f(x − h)

Horizontal shift: right h units if h > 0, left |h| units if h < 0 (opposite of sign).

−f(x) vs. f(−x)

−f(x) reflects over the x-axis; f(−x) reflects over the y-axis.

a·f(x), |a| > 1

Vertical stretch by factor a; if 0 < |a| < 1, it's a vertical compression.

f(bx), |b| > 1

Horizontal compression by factor 1/b; if 0 < |b| < 1, it's a horizontal stretch.

Try an AP-style question

How does g(x) = f(x − 3) + 2 transform f?

1. Right 3, down 2
2. Left 3, down 2
3. Right 3, up 2
4. Left 3, up 2