Dilations of Functions

A dilation stretches or compresses a graph by a scale factor from a fixed line (usually an axis). Unlike translations, which slide the graph without changing its shape, dilations actually change the distances between points, but they preserve the overall proportions of the graph.

Vertical Dilations: $y = af(x)$

When a constant is multiplied outside the function, every $y$-coordinate is multiplied by $a$. This stretches or compresses the graph vertically from the $x$-axis.

• If $a > 1$, the graph is stretched away from the $x$-axis
• If $0 < a < 1$, the graph is compressed toward the $x$-axis
• If $a < 0$, there is also a reflection in the $x$-axis

For example, $y = 2x^2$ makes the parabola narrower because every $y$-value is doubled. The point $(1, 1)$ on $y = x^2$ moves to $(1, 2)$.

Horizontal Dilations: $y = f(bx)$

When a constant is multiplied inside the function, every $x$-coordinate is divided by $b$. This stretches or compresses the graph horizontally from the $y$-axis.

• If $b > 1$, the graph is compressed toward the $y$-axis (dilation factor $\frac{1}{b}$)
• If $0 < b < 1$, the graph is stretched away from the $y$-axis (dilation factor $\frac{1}{b} > 1$)
• If $b < 0$, there is also a reflection in the $y$-axis

For example, $y = (2x)^2 = 4x^2$ compresses the parabola horizontally by factor $\frac{1}{2}$. The point $(1, 1)$ on $y = x^2$ moves to $(\frac{1}{2}, 1)$ because you need $x = \frac{1}{2}$ for $2x = 1$.

Alternative Form: $y = f\!\left(\frac{x}{b}\right)$

You will sometimes see dilations written as $f\!\left(\frac{x}{b}\right)$. In this form, the dilation factor is simply $b$ from the $y$-axis. This is often easier to read because the number in the denominator is the actual scale factor.

• $y = f\!\left(\frac{x}{3}\right)$ is a horizontal dilation by factor 3
• $y = f\!\left(\frac{x}{\frac{1}{2}}\right) = f(2x)$ is a horizontal dilation by factor $\frac{1}{2}$

Vertical dilation: $y = af(x)$, factor $a$ from the $x$-axis, $(x,y) \to (x, ay)$, $a > 1$: stretch; $0 < a < 1$: compress; Horizontal dilation: $y = f(bx)$, factor $\tfrac{1}{b}$ from the $y$-axis, $(x,y)...

Pause, copy both dilation rules with their coordinate maps: $y = af(x)$ maps $(x,y) \to (x,ay)$; $y = f(bx)$ maps $(x,y) \to (x/b, y)$ into your book.

We just saw that $y = af(x)$ scales $y$-values by $a$ and $y = f(bx)$ scales $x$-values by $\frac{1}{b}$. That raises a question: if every point shifts, how do specific features like intercepts, turning points, and asymptotes transform? This card answers it → vertical dilation leaves $x$-intercepts unchanged; horizontal dilation leaves the $y$-intercept unchanged but moves $x$-intercepts.

Dilations affect different features in specific ways:

• Vertical dilation $af(x)$:
  • $x$-intercepts stay the same (where $y = 0$)
  • $y$-intercept is multiplied by $a$
  • Range is scaled by factor $a$
  • Domain is unchanged
• Horizontal dilation $f(bx)$:
  • $y$-intercept stays the same (where $x = 0$)
  • $x$-intercepts are divided by $b$
  • Domain is scaled by factor $\frac{1}{b}$
  • Range is unchanged

Vertical dilation: $x$-intercepts unchanged; $y$-intercept multiplied by $a$; range scaled by $a$; domain unchanged; Horizontal dilation: $y$-intercept unchanged; $x$-intercepts divided by $b$; domain scaled by $\tfrac{1}{b}$; range unchanged

Pause, copy the two-column feature table: vertical dilation ($x$-intercepts fixed, $y$-intercept $\times a$) vs horizontal dilation ($y$-intercept fixed, $x$-intercepts $\div b$) into your book.