What's the Effect of a Vertical or Horizontal Stretch on a Graph?

A vertical stretch multiplies every output, or $y$-coordinate, while a horizontal stretch changes every input, or $x$-coordinate. For $y=f(qx)$, remember that the horizontal scale factor is $\frac{1}{q}$, not $q$.

This is part of Mathematics: Analysis and Approaches, SL 2.11: Transformation of functions.

How the Transformations Work

Suppose a point ((x,y)) lies on the original graph $y=f(x)$.

For a vertical transformation, the multiplication occurs outside the function, so it directly multiplies the output. If $p>1$, the graph stretches away from the $x$-axis; if $0<p<1$, it is compressed towards the $x$-axis.

For a horizontal transformation, the multiplication occurs inside the function. To obtain the same original input $x$, the transformed graph requires the coordinate $\frac{x}{q}$. Therefore, $q>1$ produces a horizontal compression, while $$ produces a horizontal stretch.

Coordinate mapping is the safest method because it works for any function, including curves that are difficult to rewrite algebraically. Points on the $y$-axis remain on the $y$-axis under a vertical stretch, while points on the $x$-axis remain fixed because multiplying $y=0$ still gives $0$. Under a horizontal stretch, points with $$ remain fixed. This gives a useful check when sketching.

For example, starting with $f(x)=x^2$:

• $y=3f(x)=3x^2$ is a vertical stretch with scale factor $3$.
• $y=f(2x)=(2x)^2=4x^2$ is a horizontal compression with scale factor .

The common misconception is that $f(2x)$ stretches the graph horizontally by factor $2$. This is incorrect: inside transformations act in the reciprocal direction. Negative values also include a reflection, not just a stretch.

Exam Technique

In an IB response, state the direction and scale factor precisely. For a sketch, transform important coordinates such as intercepts and turning points, and label key features. At SL, transformations of the combined form $f(ax+b)$ are not required; composite transformations may still be tested, and their order can matter.