Horizontal Stretch and Compression of Functions

A horizontal stretch (or compression) changes the width of a graph without changing its height. It happens whenever you replace \(x\) with \(x\) divided by a number, or with \(x\) multiplied by a number, inside a function. If \(y = f(x)\) is the original graph, then the transformed graph is written as:

\( y = f\left(\dfrac{x}{b}\right) \)

Here \(b\) controls how wide or narrow the graph becomes. Unlike a vertical stretch, which multiplies the whole output, a horizontal stretch only affects the input, so it changes the \(x\)-coordinates of every point while leaving the \(y\)-coordinates exactly where they were.

For \(y = f\left(\dfrac{x}{b}\right)\), with \(b > 0\):

• If \(b > 1\), the graph is stretched horizontally by a factor of \(b\), so it becomes wider.
• If \(0 < b < 1\), the graph is compressed horizontally, so it becomes narrower.

This same idea is often written as \(y = f(kx)\), where \(k = \dfrac{1}{b}\). In that form, \(k > 1\) compresses the graph and \(0 < k < 1\) stretches it. Both versions describe the same transformation; only the way the factor is written changes, so always check whether the number is dividing \(x\) or multiplying it before deciding whether the graph gets wider or narrower.

This is the part students find backwards at first: with a horizontal transformation, dividing \(x\) by a number bigger than 1 makes the graph wider, while multiplying \(x\) by a number bigger than 1 makes it narrower. Vertical stretches work the way you would expect (multiplying the output makes the graph taller), but horizontal transformations work inside the function, so the effect on the \(x\)-axis is the opposite of the effect on the \(y\)-axis. If the factor is negative, the stretch or compression is also flipped left to right, which is the same effect described in reflection across the y axis.

It also helps to compare this with shifting a graph sideways rather than resizing it, which is covered separately in horizontal translations.

Start with \(f(x) = x^2\). To stretch this graph horizontally by a factor of 2, replace \(x\) with \(\dfrac{x}{2}\):

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

Every point on \(f\) moves twice as far from the y axis. The point \((1, 1)\) on \(f\) becomes \((2, 1)\) on \(g\), and \((-1, 1)\) becomes \((-2, 1)\). Notice the \(y\)-values do not change at all.

Now start with \(f(x) = |x|\) and multiply the input by 3:

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

Because the input is multiplied by a number greater than 1, the graph is compressed horizontally by a factor of \(\dfrac{1}{3}\). The point \((3, 3)\) on \(f\) moves to \((1, 3)\) on \(g\): the \(x\)-coordinate shrinks by a factor of 3, while the \(y\)-coordinate stays the same.

Because a horizontal stretch or compression only changes \(x\)-coordinates, the range of the function usually stays the same, but the domain can change. For example, if the original domain of \(f\) is \([0, 4]\) and the graph is stretched by a factor of 2, the domain of \(g(x) = f\left(\dfrac{x}{2}\right)\) becomes \([0, 8]\), since every \(x\)-value doubles. If you need a refresher on describing these sets, see how to write domain and range.

• Mixing up the roles of \(b\): forgetting whether the graph is described as \(f\left(\dfrac{x}{b}\right)\) or \(f(bx)\) leads to stretching when you meant to compress, or the reverse.
• Applying the factor to the \(y\)-coordinates instead of the \(x\)-coordinates.
• Forgetting that a negative factor also reflects the graph across the y axis.
• Confusing a horizontal stretch with a horizontal shift; a stretch changes width, a shift changes position, and the two are applied differently inside the function.

Once this pattern feels comfortable, it connects well with related ideas such as 1 to 1 functions, since horizontal stretches preserve one-to-one behavior as long as the original function was one-to-one.

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