Vertical Stretch — Definition, Formula & Examples

Q: Does a vertical stretch change the x-intercepts of a function?

No. At any x-intercept, f(x) = 0, so a · f(x) = a · 0 = 0. The x-intercepts stay exactly where they are. Only points where y ≠ 0 move farther from (or closer to) the x-axis.

Vertical Stretch
Vertical Dilation

A stretch in which a plane figure is distorted vertically.

See also

Horizontal stretch

Key Formula

g(x) = a \cdot f(x), \quad a > 1

Where:

$f(x)$ = The original (parent) function
$g(x)$ = The transformed function after the vertical stretch
$a$ = The stretch factor; must be greater than 1 for a stretch (values between 0 and 1 produce a vertical compression instead)

Worked Example

Problem: The parent function is f(x) = x². Apply a vertical stretch by a factor of 3. Find the new function and evaluate it at x = −2, 0, and 2.

Step 1: Write the transformation rule. Multiply the entire output of f(x) by the stretch factor 3.

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

Step 2: Evaluate the original function at each x-value.

f(-2) = 4, \quad f(0) = 0, \quad f(2) = 4

Step 3: Evaluate the stretched function at each x-value by multiplying by 3.

g(-2) = 3(4) = 12, \quad g(0) = 3(0) = 0, \quad g(2) = 3(4) = 12

Step 4: Compare the points. Every y-coordinate has been tripled. The point (2, 4) moved to (2, 12), while (0, 0) stayed fixed because 3 × 0 = 0. The parabola is narrower and taller.

Answer: The vertically stretched function is g(x) = 3x². At x = ±2, the y-value changes from 4 to 12, and points on the x-axis remain unchanged.

Another Example

Problem: The function f(x) = sin(x) is vertically stretched by a factor of 4. Describe the new function and its amplitude.

Step 1: Apply the vertical stretch by multiplying the output by 4.

g(x) = 4\sin(x)

Step 2: The original sine function oscillates between −1 and 1. After the stretch, the new function oscillates between −4 and 4.

-4 \leq g(x) \leq 4

Step 3: The amplitude (half the distance from peak to trough) increases from 1 to 4. The period and the x-intercepts remain unchanged.

Answer: The stretched function is g(x) = 4 sin(x) with an amplitude of 4. The wave is taller but its period stays at 2π.

Frequently Asked Questions

How do you tell the difference between a vertical stretch and a vertical compression?

Both use the form g(x) = a · f(x). If a > 1, the graph is stretched (pulled away from the x-axis, making it taller). If 0 < a < 1, the graph is compressed (pushed toward the x-axis, making it shorter). The dividing line is a = 1, which leaves the graph unchanged.

Does a vertical stretch change the x-intercepts of a function?

No. At any x-intercept, f(x) = 0, so a · f(x) = a · 0 = 0. The x-intercepts stay exactly where they are. Only points where y ≠ 0 move farther from (or closer to) the x-axis.

Vertical Stretch vs. Horizontal Stretch

A vertical stretch multiplies the output: g(x) = a · f(x) with a > 1, making the graph taller. A horizontal stretch multiplies the input: g(x) = f(bx) with 0 < b < 1, making the graph wider. They affect different coordinates — vertical stretches change y-values while horizontal stretches change x-values. A common source of confusion is that the factor in a horizontal stretch works inversely: f(½x) stretches horizontally by a factor of 2.

Why It Matters

Vertical stretches appear throughout algebra, trigonometry, and physics whenever you scale a quantity. Changing the amplitude of a sound wave or adjusting the intensity of a signal are real-world vertical stretches. Understanding this transformation also helps you quickly sketch graphs of function families without plotting point by point.

Common Mistakes

Mistake: Confusing a vertical stretch with a horizontal compression. Students see g(x) = 3x² and think the parabola is 'narrower because of a horizontal change.'

Correction: The graph looks narrower, but the transformation is vertical: every y-value is tripled. Recognizing whether the factor multiplies the output (vertical) or the input (horizontal) is the key distinction.

Mistake: Applying the stretch factor to x-coordinates instead of y-coordinates.

Correction: In g(x) = a · f(x), only the y-values are multiplied by a. The x-coordinates of each point remain the same. To change x-coordinates, you need a horizontal transformation inside the function's argument.

Related Terms

Dilation — General scaling transformation that includes stretches
Horizontal Stretch — Stretch that scales x-values instead of y-values
Vertical — Direction along which the stretch occurs
Plane Figure — Two-dimensional shape that can be stretched
Transformation — Broad category including stretches and shifts
Parent Function — The original function before any transformation
Amplitude — Vertical stretch changes amplitude in trig functions