Dilation of a Graph

Q: What is the difference between dilation and compression of a graph?

Dilation (stretch) and compression are actually the same type of transformation — they differ only in the size of the scale factor. When the factor is greater than 1, the graph stretches away from an axis (dilation). When the factor is between 0 and 1, the graph squeezes toward an axis (compression). Some textbooks use 'dilation' as the umbrella term for both.

Q: Why does horizontal dilation use x divided by the factor instead of x times the factor?

Horizontal transformations work opposite to what you might expect because the factor acts on the input. To stretch a graph horizontally by factor b, each output value must be reached at an x-value that is b times farther from the origin. Replacing x with x/b achieves this: the point that was at x = c on f now appears at x = bc on g. This inverse relationship is why horizontal dilation divides by b rather than multiplying.

Q: How do you tell if a graph has been vertically or horizontally dilated?

Look at how the shape changes relative to the axes. A vertical dilation changes how tall the graph is — points move farther from or closer to the x-axis. A horizontal dilation changes how wide the graph is — points move farther from or closer to the y-axis. If the function's formula has a constant multiplied outside, it is vertical; if the constant is applied to x inside the function, it is horizontal.

Stretch
Dilation of a Graph

A transformation in which all distances on the coordinate plane are lengthened by multiplying either all x-coordinates (horizontal dilation) or all y-coordinates (vertical dilation) by a common factor greater than 1.

Note: When the common factor is less than 1 the transformation is called a compression.

Two sine curves on coordinate axes: original graph (normal amplitude) and same graph vertically dilated (increased amplitude).

Key Formula

\text{Vertical dilation: } g(x) = a \cdot f(x) \qquad \text{Horizontal dilation: } g(x) = f\!\left(\frac{x}{b}\right)

Where:

$f(x)$ = The original function
$g(x)$ = The transformed (dilated) function
$a$ = Vertical scale factor; |a| > 1 stretches vertically, 0 < |a| < 1 compresses vertically
$b$ = Horizontal scale factor; |b| > 1 stretches horizontally, 0 < |b| < 1 compresses horizontally

Worked Example

Problem: Given f(x) = x², find the equation of the graph after a vertical dilation by a factor of 3. Then find the new y-coordinate when x = 2.

Step 1: Identify the type of dilation. A vertical dilation by factor 3 means every y-value is multiplied by 3.

g(x) = 3 \cdot f(x)

Step 2: Substitute the original function into the formula.

g(x) = 3 \cdot x^2 = 3x^2

Step 3: Evaluate the original function at x = 2.

f(2) = 2^2 = 4

Step 4: Evaluate the dilated function at x = 2.

g(2) = 3(2^2) = 3(4) = 12

Answer: The dilated graph is g(x) = 3x². At x = 2, the y-coordinate changes from 4 to 12, confirming the graph is stretched vertically by a factor of 3.

Another Example

This example demonstrates horizontal dilation, which involves replacing x with x/b inside the function — a common source of confusion because the factor appears in the denominator, unlike vertical dilation.

Problem: Given f(x) = x², find the equation of the graph after a horizontal dilation by a factor of 2. Then find the new y-coordinate when x = 6.

Step 1: Identify the type of dilation. A horizontal dilation by factor 2 means every x-coordinate is stretched to twice its original distance from the y-axis. In the formula, you replace x with x/b where b = 2.

g(x) = f\!\left(\frac{x}{2}\right)

Step 2: Substitute the original function.

g(x) = \left(\frac{x}{2}\right)^2 = \frac{x^2}{4}

Step 3: Check: The point (3, 9) on the original graph should map to (6, 9) on the dilated graph, since horizontal distances double. Evaluate g(6).

g(6) = \frac{6^2}{4} = \frac{36}{4} = 9 \checkmark

Step 4: Compare: On the original, x = 6 gives f(6) = 36. On the dilated graph, g(6) = 9. The graph is wider — it has been stretched horizontally.

f(6) = 36 \quad \text{vs.} \quad g(6) = 9

Answer: The horizontally dilated graph is g(x) = x²/4. At x = 6, y = 9. The parabola is wider because it has been stretched horizontally by a factor of 2.

Frequently Asked Questions

What is the difference between dilation and compression of a graph?

Dilation (stretch) and compression are actually the same type of transformation — they differ only in the size of the scale factor. When the factor is greater than 1, the graph stretches away from an axis (dilation). When the factor is between 0 and 1, the graph squeezes toward an axis (compression). Some textbooks use 'dilation' as the umbrella term for both.

Why does horizontal dilation use x divided by the factor instead of x times the factor?

Horizontal transformations work opposite to what you might expect because the factor acts on the input. To stretch a graph horizontally by factor b, each output value must be reached at an x-value that is b times farther from the origin. Replacing x with x/b achieves this: the point that was at x = c on f now appears at x = bc on g. This inverse relationship is why horizontal dilation divides by b rather than multiplying.

How do you tell if a graph has been vertically or horizontally dilated?

Look at how the shape changes relative to the axes. A vertical dilation changes how tall the graph is — points move farther from or closer to the x-axis. A horizontal dilation changes how wide the graph is — points move farther from or closer to the y-axis. If the function's formula has a constant multiplied outside, it is vertical; if the constant is applied to x inside the function, it is horizontal.

Vertical Dilation vs. Horizontal Dilation

	Vertical Dilation	Horizontal Dilation
Formula	g(x) = a · f(x)	g(x) = f(x / b)
Where the factor appears	Outside the function (multiplies output)	Inside the function (divides the input)
Effect when factor > 1	Graph stretches away from x-axis (taller)	Graph stretches away from y-axis (wider)
Effect when 0 < factor < 1	Graph compresses toward x-axis (shorter)	Graph compresses toward y-axis (narrower)
Which coordinates change	All y-coordinates are multiplied by a	All x-coordinates are multiplied by b
Fixed axis	Points on the x-axis stay fixed (y = 0)	Points on the y-axis stay fixed (x = 0)

Why It Matters

Dilation of a graph appears frequently in algebra, precalculus, and physics whenever you need to scale a function — for example, adjusting the amplitude of a sine wave or changing units on a model. Understanding dilation helps you quickly sketch transformed graphs without plotting point by point. It is also essential for interpreting real-world models where a parameter controls the rate or magnitude of change, such as doubling a spring constant or scaling a probability distribution.

Common Mistakes

Mistake: Multiplying x by the horizontal scale factor instead of dividing by it.

Correction: For a horizontal stretch by factor b, use g(x) = f(x/b), not f(bx). Replacing x with bx actually compresses the graph horizontally by factor b (the opposite effect). Remember: horizontal transformations act in the 'opposite' direction from what the number suggests.

Mistake: Assuming dilation changes the shape of the graph rather than just its scale.

Correction: A dilation preserves the overall shape of the curve — a parabola stays a parabola, a sine wave stays a sine wave. Only the scale (height or width) changes. If the shape looks fundamentally different, you may have applied the transformation incorrectly.

Related Terms

Transformations — Dilation is one of the four main graph transformations
Coordinate Plane — The setting where graph dilations are performed
Horizontal Stretch — A horizontal dilation with factor greater than 1
Vertical Stretch — A vertical dilation with factor greater than 1
Compression — A dilation with factor between 0 and 1
Dilation — General term for scaling transformations
Dilation of a Geometric Figure — Dilation applied to shapes rather than function graphs