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

In particular, a translation changed the location of an object or function in the coordinate plane but left all other attributes the same.

In this module we will examine specific types of reflections:

• reflections in the \(x\)-axis,
• reflections in the \(y\)-axis, and
• reflections in both the \(x\)-axis and the \(y\)-axis.

On the graph, \(\triangle ABC\) is reflected in the \(y\)-axis. The image \(\triangle A'B'C'\) is shown.

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

The size and shape of an object or function is not altered by a reflection.

The location and orientation are 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 move in a counter-clockwise direction.

The line of reflection is a right bisector of a line segment connecting any pre-image point to its corresponding image point.

In our example, the line of reflection is the \(y\)-axis.

We note that it is is perpendicular to \(AA'\), \(BB'\), and \(CC'\).

Also, \(AA'\), \(BB'\), and \(CC'\) intersect the \(y\)-axis at \(D\), \(E\), and \(F\), respectively, such that \(AD=DA'\), \(BE=EB'\) and \(CF=FC'\).

Any point on the line of reflection is unaffected by the reflection. These points are said to be invariant. In our example, there are no invariant points.

Consider some function \(y=f(x)\).

• If \(g(x)=-f(x)\), how are the graphs of \(f(x)\) and \(g(x)\) related?
• If \(h(x)=f(-x)\), how are the graphs of \(f(x)\) and \(h(x)\) related?
• If \(k(x)=-f(-x)\), how are the graphs of \(f(x)\) and \(k(x)\) related?

To investigate these questions, use the following worksheet.

When you have completed the investigation, you should be able to answer the above questions.