Mathwords logoMathwords

Horizontal Stretch — Definition, Formula & Examples

Horizontal Stretch
Horizontal Dilation

A stretch in which a plane figure is distorted horizontally.

 

 

See also

Vertical stretch

Key Formula

y=f(xk),k>1y = f\left(\frac{x}{k}\right), \quad k > 1
Where:
  • f(x)f(x) = The original (parent) function
  • kk = The horizontal stretch factor; when k > 1 the graph stretches wider
  • xx = The input variable

Worked Example

Problem: The parent function is f(x) = x². Apply a horizontal stretch by a factor of 3 and find the new function. Then verify using the point (3, 9) from the original graph.
Step 1: Write the horizontal stretch formula. Replace x with x/k, where k = 3.
g(x)=f ⁣(x3)=(x3)2=x29g(x) = f\!\left(\frac{x}{3}\right) = \left(\frac{x}{3}\right)^{2} = \frac{x^{2}}{9}
Step 2: Take the original point (3, 9). Under a horizontal stretch by factor 3, the x-coordinate is multiplied by 3 while the y-coordinate stays the same.
(3×3,  9)=(9,  9)(3 \times 3,\; 9) = (9,\; 9)
Step 3: Verify: plug x = 9 into the new function to confirm the y-value is still 9.
g(9)=929=819=9  ✓g(9) = \frac{9^{2}}{9} = \frac{81}{9} = 9 \; \checkmark
Answer: The horizontally stretched function is g(x) = x²/9. The point (3, 9) on the original parabola moves to (9, 9) on the stretched graph, confirming the graph is 3 times wider.

Another Example

Problem: The function f(x) = |x| is horizontally stretched by a factor of 2. Write the new function and describe how the graph changes.
Step 1: Apply the horizontal stretch formula with k = 2.
g(x)=f ⁣(x2)=∣x2∣g(x) = f\!\left(\frac{x}{2}\right) = \left|\frac{x}{2}\right|
Step 2: Check the original point (4, 4). Multiply the x-coordinate by 2.
(4×2,  4)=(8,  4)(4 \times 2,\; 4) = (8,\; 4)
Step 3: Verify with the new function.
g(8)=∣82∣=4  ✓g(8) = \left|\frac{8}{2}\right| = 4 \; \checkmark
Answer: The new function is g(x) = |x/2|. The V-shape of the absolute value graph becomes twice as wide, opening more gradually.

Frequently Asked Questions

Why do you divide x by the stretch factor instead of multiplying?
A horizontal stretch by factor k means each point's x-coordinate is multiplied by k. To make the function produce the same y-value at the new, larger x, you must divide the input by k inside the function. This is the opposite of what intuition suggests: multiplying inside the function would actually compress the graph horizontally.
What is the difference between a horizontal stretch and a horizontal compression?
Both use the form f(x/k). When k > 1, the graph stretches wider (horizontal stretch). When 0 < k < 1, the graph squeezes narrower (horizontal compression). Equivalently, if you see f(bx) with 0 < b < 1, that is a stretch by factor 1/b, and with b > 1, that is a compression by factor 1/b.

Horizontal Stretch vs. Vertical Stretch

A horizontal stretch changes x-coordinates by replacing x with x/k inside the function, making the graph wider. A vertical stretch changes y-coordinates by multiplying the entire function by a factor, making the graph taller: y = k · f(x). Horizontal transformations act on the input; vertical transformations act on the output.

Why It Matters

Horizontal stretches appear whenever you need to model changes in the rate or period of a process. For example, stretching a sine wave horizontally changes its period, which is essential in modeling sound waves, tides, and alternating current. Understanding this transformation also helps you quickly sketch graphs of unfamiliar functions by relating them to simpler parent functions.

Common Mistakes

Mistake: Multiplying x by the stretch factor inside the function instead of dividing.
Correction: Remember that horizontal transformations work opposite to what you might expect. To stretch horizontally by factor k, replace x with x/k. Multiplying x by k inside the function actually compresses the graph.
Mistake: Confusing a horizontal stretch with a vertical stretch because both make the graph 'look different.'
Correction: Ask yourself: are x-values changing or y-values changing? If every point moves left or right (x changes), it is horizontal. If every point moves up or down (y changes), it is vertical. A horizontal stretch of x² gives x²/9; a vertical stretch of x² gives 9x². These are different functions with different shapes.

Related Terms