Vertical Stretch and Compression of Functions

A vertical stretch is a transformation that pulls a graph away from the x axis, making it taller. A vertical compression (also called a vertical shrink) does the opposite: it pushes a graph closer to the x axis, making it flatter. Both transformations come from the same simple rule, so it helps to study them together.

If \(y = f(x)\) is the original function, then a vertical stretch or compression is written as:

\(y = a \cdot f(x)\)

Here \(a\) is a positive number that scales every output value of the function. Only the y-coordinates change, so a point \((x, y)\) on the graph of \(f(x)\) moves to \((x, a y)\) on the graph of \(a f(x)\).

• If \(a > 1\), the graph is stretched vertically (it gets taller).
• If \(0 < a < 1\), the graph is compressed vertically (it gets flatter).
• If \(a = 1\), there is no change, the graph is the original function.
• If \(a < 0\), the graph is stretched or compressed by \(|a|\) and also reflected across the x axis.

That last case connects directly to reflection across the x axis, since multiplying by a negative number flips the graph upside down at the same time it rescales it.

Start with the base function \(f(x) = x^2\). To stretch it vertically by a factor of 2, multiply the whole function by 2:

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

Compare a few points:

Every y-value from the original function has doubled, while the x-values, and therefore the vertex at the origin, stay exactly where they were. Here is the stretched graph, using \(g(x) = 3x^2\) to show an even taller example clearly:

Notice the vertex still sits at \((0, 0)\), but every other point on the parabola is now three times as far from the x axis as it was on the original curve.

Now take \(f(x) = |x|\) and compress it vertically by a factor of \(\frac{1}{4}\):

\(h(x) = \frac{1}{4}|x|\)

Since \(0 < \frac{1}{4} < 1\), every output is shrunk toward the x axis, making the familiar V-shape look much flatter and wider without moving it left, right, up, or down.

Because a vertical stretch or compression only rescales y-values, the set of allowed x-values never changes. If you already know how to write domain and range for the base function, you can reuse the same domain for the transformed function. The range, however, usually does change, since every output has been multiplied by \(a\).

It is easy to confuse a vertical stretch with a shift. A horizontal translation slides a graph left or right by changing the input, while a vertical stretch rescales the output and keeps every x-intercept in the same horizontal position. Recognizing which part of the function is being changed, the input inside the parentheses or the output multiplied outside, is the key to telling these transformations apart.

When two or more transformations appear in the same equation, such as \(y = -2f(x)\), work through them one effect at a time: apply the stretch factor of 2 first, then apply the reflection caused by the negative sign.

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