In a previous module, we noted that a transformation changes one or more of the location, shape, size, or orientation of an object in the coordinate plane.

A translation changed the location of an object or function in the coordinate plane but left all other attributes the same.

A reflection in the \(x\)-axis or \(y\)-axis changed the orientation and location of the object or function in the coordinate plane but left all other attributes the same.

In this module, we examine

• horizontal stretches about the \(y\)-axis,
• vertical stretches about the \(x\)-axis, and
• stretches about both the \(x\)-axis and the \(y\)-axis.

On the graph, \(\triangle ABC\) is stretched vertically about the \(x\)-axis by a factor of \(2\).

The image \(\triangle A'B'C'\) is shown.

We will use this specific diagram to make some observations about stretches.

The size of an object or shape of a function is changed as a result of a stretch.

The orientation is not changed.

On the pre-image, from \(A\) to \(B\) to \(C\), we move in a clockwise direction.

On the image, from \(A'\) to \(B'\) to \(C'\), we also move in a clockwise direction.

The perpendicular distance from the \(x\)-axis to any point on the image is twice as far as the perpendicular distance from the \(x\)-axis to the corresponding point on the pre-image.

On the diagram, \(\lvert PB'\rvert=2\times \lvert PB\rvert\) and \(\lvert QC'\rvert=2\times \lvert QC\rvert\).

The absolute value bars are used to indicate the length of each line segment.

Any point on the \(x\)-axis will be invariant as a result of a vertical stretch about the \(x\)-axis.

In this case, \(A(1,0)\) is invariant.

Consider the function \(y=f(x)\). Five specific points on the function are shown. Sketch the graph \(g(x)=3f(x)\).

Solution

Using a table of values, we will determine the image points associated with the five specific points on \(f(x)\).

The image is shown on the graph.

Comparing the image points to the corresponding pre-image points, note that the \(x\)-coordinates are unchanged and the \(y\)-coordinates are \(3\) times their original value.

To obtain an image point, the mapping is \((x,y)\rightarrow (x,3y)\).

Points \(B\) and \(D\) are invariant since they are on the \(x\)-axis.

We would say that \(y=f(x)\) is stretched vertically about the \(x\)-axis by a factor of \(3\).

Consider the function \(y=f(x)\). Five specific points on the function are shown. Sketch the graph \(g(x)=\dfrac{1}{2}f(x)\).

Solution

Using a table of values, we will determine the image points associated with the five specific points on \(y=f(x)\).

The image is shown on the graph.

Comparing the image points to the corresponding pre-image points, note that the \(x\)-coordinates are unchanged and the \(y\)-coordinates are half of their original value.

To obtain an image point, the mapping is \((x,y)\rightarrow \left(x,\frac{1}{2}y\right)\). Points \(B\) and \(D\) are invariant since they are on the \(x\)-axis.

We would say that \(y=f(x)\) is stretched vertically about the \(x\)-axis by a factor of \(\frac{1}{2}\).

Can we make some general conclusions concerning vertical stretches about the \(x\)-axis?

Use the worksheet to further investigate vertical stretches about the \(x\)-axis.

Using a slider to change the values of \(a\) in \(y=af(x)\), you will be able to stretch our five base functions.

When you have completed the investigation, you should be able to make conclusions about comparing \(y=af(x)\) to \(y=f(x)\) for values of \(a\gt 1\) and for values of \(a\) such that \(0\lt a \lt 1\).