Lesson 5: Compressions and Stretches of Functions

Property

When a function is multiplied by $-1$, the graph reflects across the x-axis:

Every point $(x, y)$ on the original graph becomes $(x, -y)$ on the reflected graph.

Property

For any function $f(x)$, transformations follow these patterns:

• Vertical: $g(x) = af(x)$ where $|a| > 1$ stretches, $0 < |a| < 1$ compresses
• Horizontal: $g(x) = f(bx)$ where $|b| > 1$ compresses, $0 < |b| < 1$ stretches
• Reflection: $g(x) = -f(x)$ reflects across x-axis

Examples

Property

Vertical transformations modify the output: $g(x) = k \cdot f(x)$

Horizontal transformations modify the input: $g(x) = f(k \cdot x)$

Property

The coefficient $a$ in the function $f(x) = ax^2$ affects the graph of $f(x) = x^2$ by stretching or compressing it.

• If $0 < |a| < 1$, the graph of $f(x) = ax^2$ will be “wider” than the graph of $f(x) = x^2$.
• If $|a| > 1$, the graph of $f(x) = ax^2$ will be “skinnier” than the graph of $f(x) = x^2$.

Examples

• The graph of $f(x) = 4x^2$ is skinnier than $f(x) = x^2$. For any given x-value, the y-value is multiplied by 4, stretching the parabola vertically.

• The graph of $f(x) = \frac{1}{3}x^2$ is wider than $f(x) = x^2$. The y-values are compressed to one-third of their original height, making the parabola open more broadly.