Vertical Dilation

Scaling the Heights of a Function Graph

Vertical dilation changes how tall a function graph looks without moving any of its horizontal positions. Stretching the graph makes it taller, and compressing it brings the graph closer to the $x$-axis, while every point keeps its original $x$ value.

If you have a function $f(x)$, then vertical dilation produces a new function $g(x) = k \cdot f(x)$. Here $k \ne 0$ is the factor that multiplies each output value.

How the Output Factor Changes the Height

For any function $f(x)$ and a nonzero factor $k$, vertical dilation is defined as:

The factor $k$ decides what happens to the height:

• If $k > 1$, the graph is stretched vertically (enlarged)
• If $0 < k < 1$, the graph is compressed vertically (reduced)
• If $k = 1$, the graph does not change
• If $k < 0$, the graph undergoes reflection as well as dilation

The value $k = 0$ is excluded because it sends every output to $0$. The whole graph then collapses onto the $x$-axis.

Here the factor works in the direction you expect: a factor greater than one makes the graph taller, and a factor between zero and one makes the graph shorter.

Visualization of Vertical Dilation

The quadratic function $f(x) = x^2$ shows how several scale factors change vertical distance from the $x$-axis.

The graph shows:

• The original function $f(x) = x^2$ as reference
• Function $g(x) = 2x^2$ is vertically stretched by factor $2$
• Function $h(x) = 0.5x^2$ is vertically compressed by factor $0.5$
• All graphs have the same vertex at $(0, 0)$

Vertical Dilation on Linear Functions

The same rule applies to the linear function $f(x) = x + 1$.

The three lines show:

• The original function $f(x) = x + 1$ has slope $1$
• Function $g(x) = 3(x + 1)$ has slope $3$ (stretched)
• Function $h(x) = 0.5(x + 1)$ has slope $0.5$ (compressed)
• All lines still intersect the $y$-axis, but at different points

Which Values the Vertical Scale Moves

Once you know whether the graph grows taller or shorter, you still need to know which values move. Vertical dilation multiplies every output value by the factor $k$, so heights change while positions along the $x$-axis stay the same.

Take a point $(a, b)$ on the graph of $f(x)$. The matching point on the graph of $k \cdot f(x)$ is $(a, k \cdot b)$. The $x$ coordinate stays at $a$ because vertical dilation never moves a point sideways.

The range grows or shrinks together with the graph, because it lists the output values. The domain keeps its own values because the allowed inputs do not change. A horizontal dilation divides the horizontal coordinates, so the heights stay unchanged.

Scaling the Values of an Exponential Function

For the exponential function $f(x) = 2^x$, the scale factor multiplies every function value.

For exponential functions:

• The horizontal asymptote remains at $y = 0$ for both functions
• The $y$-intercept changes from $(0, 1)$ to $(0, 2)$
• Every function value doubles, while the exponential base and relative growth rate stay the same

Vertical Dilation with Negative Factor

A negative scale factor combines vertical scaling with reflection across the $x$-axis.

When the scale factor is negative:

• The graph undergoes reflection across the $x$-axis
• Simultaneously undergoes dilation according to the absolute value of the scale factor
• A parabola that opens upward becomes one that opens downward

Exercises

The three problems below use the height rule you have just learned. Each one multiplies the output by a stated factor, so watch whether that factor is greater than one, between zero and one, or negative. Try them without the answer key first, then compare your steps with the worked answers.

1. Given the function $f(x) = x^2 - 4x + 3$. Determine the equation of the function resulting from vertical dilation with scale factor $3$.

2. The graph of the function $g(x) = \sqrt{x}$ undergoes vertical dilation with factor $\frac{1}{2}$. Determine the equation of the resulting function, then determine the range of that function.

3. Function $h(x) = |x - 1| + 2$ undergoes vertical dilation with factor $-1$. Determine the vertex of the resulting dilated function.

Answer Key

Every worked answer starts by multiplying the whole output expression by the factor, because that is where the height rule lives. The equation is then simplified and the new graph is checked to confirm whether it grew taller or shorter.

1. Vertical dilation with factor $3$: $f'(x) = 3f(x) = 3(x^2 - 4x + 3) = 3x^2 - 12x + 9$

   Function $f(x) = x^2 - 4x + 3$ and Its Dilation Result

   The original parabola is vertically stretched by factor $3$ producing a taller parabola.

   GridAutomatic rotation

2. Equation of the resulting dilated function:

   • Vertical dilation: $g'(x) = \frac{1}{2}g(x) = \frac{1}{2}\sqrt{x}$
   • Range after dilation: $[0, \infty)$ remains $[0, \infty)$. At each input, the positive function value is halved

   Visualization:

   Function $g(x) = \sqrt{x}$ and Its Dilation Result

   The square root curve is vertically compressed by factor $0.5$ producing a lower curve.

   GridAutomatic rotation

3. The original function $h(x) = |x - 1| + 2$ has its vertex at $(1, 2)$. After vertical dilation with factor $-1$: $h'(x) = -1 \cdot (|x - 1| + 2) = -|x - 1| - 2$, the vertex becomes $(1, -2)$.

   Function $h(x) = |x - 1| + 2$ and Its Dilation Result

   Factor $-1$ reflects the absolute value function across the $x$-axis without changing its vertical scale.

   GridAutomatic rotation