Vertical Stretching and Compression (Scaling) of Graphs

This interactive tool helps you explore and understand the stretching and compression of the graph of a function when this function is multiplied by a constant \(a\). The function to be analyzed is of the form \(a \cdot f(x)\).

Select Function

Vertical Scaling Factor: \(a = \) 1.0

-3 1.0 3

Function Information

Original Function: \(f(x) = x^2\)

\(a \cdot f(x) = 1.0 \cdot x^2\)

When \(a > 1\): Vertical stretching
When \(0 < a < 1\): Vertical compression
When \(a < 0\): Reflection about the x-axis plus stretching/compression

Original Function \(f(x)\)

Scaled Function \(a \cdot f(x)\)

Graph Visualization

Tutorial

How does the multiplication of a function by a constant \(a\) affect the graph of this function?

Click on any function button above to select it.
Use the slider to set the constant \(a\) to different values and observe the effect on the graph.
Answer the following questions:

Question 1: What is the range of values of the constant \(a\) that create a vertical compression?

Question 2: What is the range of values of the constant \(a\) that create a vertical stretching?

Question 3: What values of \(a\) reflect the graph on the x-axis?

Question 4: What happens when \(a = 1\)? What about when \(a = 0\)?

Explain analytically: For a function \(f(x)\), the transformed function \(a \cdot f(x)\):

Vertically stretches the graph by a factor of \(|a|\) if \(|a| > 1\)
Vertically compresses the graph by a factor of \(|a|\) if \(0 < |a| < 1\)
Reflects the graph about the x-axis if \(a < 0\)
Leaves the graph unchanged if \(a = 1\)
Collapses the graph to the x-axis (becomes \(y = 0\)) if \(a = 0\)