Function Transformations: Shifts, Reflections, and Stretches

Function transformations let you predict a graph from a familiar parent function. Instead of plotting many points from scratch, you can understand how each parameter moves or reshapes the original curve.

The general transformation form

A useful model is y=a·f(b(x-h))+k.

• h shifts the graph horizontally.
• k shifts it vertically.
• a controls vertical stretch, compression, and reflection.
• b controls horizontal stretch, compression, and reflection.

Horizontal and vertical shifts

Starting from f(x)=x^2:

• (x-3)^2 shifts the graph 3 units right.
• (x+3)^2 shifts it 3 units left.
• x^2+2 shifts it 2 units up.
• x^2-2 shifts it 2 units down.

The sign inside the parentheses appears opposite to the direction of the horizontal shift. This is one of the most common sources of error.

Reflections

Multiplying the whole function by -1 reflects the graph across the x-axis: y=-f(x). Replacing x by -x reflects the graph across the y-axis: y=f(-x).

Vertical stretches and compressions

For y=a·f(x):

• If |a|>1, the graph is stretched vertically.
• If 0<|a|<1, the graph is compressed vertically.
• If a<0, a reflection across the x-axis is included.

Horizontal stretches and compressions

For y=f(bx), the horizontal scale factor is 1/|b|. Thus f(2x) is horizontally compressed, while f(x/2) is horizontally stretched.

Worked comparison

Graph these three functions together:

1. y=x^2
2. y=(x-2)^2-1
3. y=-2*(x-2)^2-1

The second graph shifts the vertex from (0,0) to (2,-1). The third has the same vertex, reflects across a horizontal line through the vertex, and becomes narrower because of the factor 2.

Use parent functions

The same ideas apply to sqrt(x), abs(x), 1/x, exponential functions, logarithms, and trigonometric functions. Graph the parent and transformed versions together using distinct colors.

A transformation checklist

1. Identify the parent function.
2. Read horizontal changes inside the input.
3. Read vertical changes outside the function.
4. Apply reflections.
5. Apply stretches or compressions.
6. Verify key points and asymptotes.

Open the Graphing Calculator and compare transformed functions →

For rational and exponential graphs, the next useful topic is finding vertical, horizontal, and oblique asymptotes.