How to Scale a Function Vertically?

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

Related Topics

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

Step by step guide to vertical scaling

There are four types of transformation possible for a graph of a function, which are:

• Rotations
• Translations
• Reflections
• Scaling

In addition, scaling can be divided into two different types, e.g.

• Horizontal scaling
• Vertical scaling

Vertical scaling refers to changing the shape and size of a function graph along the \(y\)-axis and is done by multiplying the function by a fixed value.

For example: We have a function \(y=f(x)\)

We have multiplied the function by \(2\), that is: \(y=2 f(x)\)

The distance of the points on the curve becomes farther from the \(x\)-axis.

The shape of the curve depends on the value of \(C\):

• If \(C > 1\), the graph stretches and makes the graph steeper.
• If \(C < 1\), the graph shrinks and makes the graph flatter.

How to do vertical scaling?

Let’s understand this with an example:

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

If we want to vertically scale this chart, we have to follow the given steps:

Step 1: Select the constant with which we want to scale the function.

Here we have selected \(+2\).

Step 2: Write the new function as \(g(x)=C f(x)\), where \(C\) is the constant.

Here, the new function will be: \(g(x)=2 f(x)= 2 x^2\)

Step 3: Trace the new function graph by replacing each value of \(y\) with \(Cy\).

Here we need to replace the value of the \(y\)-coordinate by \(2y\).

The \(Y\) coordinates of each point in the graph are multiplied by \(±C\), and the curve is shrinks or stretches accordingly.

Here we have the graph \(x\) and it is stretched in the \(y\)-direction with a factor of \(+2\).

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

Vertical Scaling – Example 1:

Vertically stretch the function \(y=(x+2)\) by a factor of two.

Exercises for Vertical Scaling

• Vertically stretch the function \(f(x)=x^3\) by a factor of \(-\frac{1}{3}\).

Vertically stretch the function \(f(x)=sin x\) by a factor of \(3\).

Vertically stretch the function \(f(x)=x^3\) by a factor of \(-\frac{1}{3}\).

Vertically stretch the function \(f(x)=sin x\) by a factor of \(3\).