Transformations

Objective

Predict how changes to a function's equation shift, stretch, or reflect its graph.

What it means

A transformation changes the position or shape of a parent function's graph. For y = a·f(x - h) + k: h shifts horizontally, k shifts vertically, a stretches/compresses vertically, and a negative a reflects over the x-axis.

Key terms

Vertical shift

Adding k outside the function moves the graph up (k>0) or down (k<0).

Horizontal shift

Replacing x with (x - h) moves the graph right (h>0) or left (h<0).

Reflection

A negative sign flips the graph over an axis.

Rules to remember

• y = f(x) + k shifts up k units (down if k is negative).
• y = f(x - h) shifts right h units (left if h is negative).
• y = a·f(x) stretches vertically if |a| > 1, compresses if 0 < |a| < 1.
• y = -f(x) reflects over the x-axis; y = f(-x) reflects over the y-axis.

Step by step

1. 1Find hLook inside the function's parentheses for (x - h).
2. 2Find kLook for a constant added or subtracted outside the function.
3. 3Find aLook for a coefficient multiplying the function.
4. 4Describe the transformationCombine shift, stretch, and reflection in order.