How to Scale a Function Horizontally?

Horizontal scaling refers to the shrinking or stretching of the curve along the \(x\)-axis by some specific units.

Related Topics

• Transformation: Rotations, Reflections, and Translations
• Transformation: Reflection

Step by step guide to horizontal scaling

Horizontal scaling means stretching or shrinking the diagram of function along the \(x\)-axis. The horizontal scale can be done by multiplying the input by a constant.

Consider the following example:

If we have a function,\(y=f(x)\), the horizontal scaling of this function can be written as: \(y=f(Cx)\).

Note:

• If \(C>1\), the graph shinks.
• If \(C<1\), the graph stretches.

The points related to the curves can be related from the following table:

How is a graph scaled horizontally?

• Steps 1: Select a constant with which we want to scale the function.
• Steps 2: Write the new function as \(g(x)=fC(x)\), where \(C\) is the constant.
• Steps 3: Trace the new function graph by replacing each value of \(x\) with \(Cx\).
• Steps 4: \(X\) coordinate of each point in the graph is multiplied by \(±C\), and the curve shrinks or stretches accordingly.

Let’s understand this with an example:

Suppose we have a quadratic equation \(f(x)=x^2\) and the graphical representation of the diagram is shown below:

We want to scale this function by a factor of \(+2\). So the equation of the new function will be:

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

So we have to plot a function:\(f(x)=(2x)^2\), and it is shrunk by a factor of \(+2\) units in the \(x\)-direction.

Note: as we have scaled it with a factor of \(+2\) units, it has made the graph steeper.

Horizontal Scaling – Example 1:

Horizontally stretch the function \(f(x)=x+2\) by a factor of \(2\) units.

Exercises for Horizontal Scaling

• Horizontal scaling of function \(f(x) = sin x\) by a factor of \(-3\).

• Horizontal scaling of function \(f(x) =x^2+3x+2\) by a factor of \(4\).

Horizontal scaling of function \(f(x) = sin x\) by a factor of \(-3\).

Horizontal scaling of function \(f(x) =x^2+3x+2\) by a factor of \(4\).