Lesson 2.4: Function Transformations (Shift & Scale)

Shifts and Scaling

Describe vertical/horizontal shifts and vertical/horizontal scaling. Predict and construct transformed graphs.

Learning Objectives

Given $y=2(x+3)^2-1$ from $y=x^2$

1) Horizontal compression by 1/2 ( $2x$ )

2) Shift left 3 ( $x+3$ )

3) Vertical stretch by 2 ( $2\cdot$ )

4) Shift down 1 ( $-1$ )

Starting from $y=f(x)$ , describe $y=-3f(x-2)+4$ .

Right 2 (due to $x-2$ ), vertical reflection across x-axis and stretch by 3 (factor -3), then up 4.

Map $y=x^2$ to $y=\tfrac{1}{2}(x+4)^2-9$ via steps.

Left 4; vertical compress by 1/2; down 9.

Compare $y=3f(x)$ and $y=f(3x)$ effects on graph width/height.

$3f(x)$ changes height (vertical); $f(3x)$ compresses horizontally by factor 1/3.

Daily temperature model $T(t)=15+10sin( frac{pi}{12}(t-8))$ , apply shift to represent a heatwave +3°C.

New model: $T_*(t)=T(t)+3=18+10sin( frac{pi}{12}(t-8))$

Sign confusion: $f(x+h)$ shifts left by h, not right.
Order matters when composing transformations; apply inner (x changes) before outer (y changes).
Mistaking horizontal scaling factor: $f(kx)$ compresses by $1/k$ , not k.

1) Describe steps to obtain $y=-2(x-1)^2+7$ from $y=x^2$

Solution

Right 1; vertical stretch by 2; reflect over x-axis; up 7

2) Compare $y=f(x)+3$ and $y=f(x+3)$ on the same axes

Solution

First shifts up 3; second shifts left 3

3) For $y=\tfrac{1}{3}f(2x-4)-5$ , describe the sequence of transformations.

Solution

Right 2 (solve 2x-4=0); horizontal compression by 1/2; vertical compression by 1/3; down 5