A transformation changes one or more of the location, shape, size, or orientation of an object in the coordinate plane.

The object could be, for example, a function like \(y=x^2\), or a geometric shape, like \(\triangle ABD\), shown below.

The initial object is called the pre-image.

The object that results after transforming the pre-image is called the image.

When moving from \(A\) to \(B\) to \(D\) in \(\triangle ABD\), we would say that there is a clockwise orientation.

Let's create an image by sliding \(\triangle ABD\) \(2\) units to the left and \(1\) unit down.

A transformation that changes only the location of the object in the plane is called a translation.

The size, shape, and orientation of \(\triangle ABD\) would not be changed as a result of sliding the triangle to the left and down.

The point that corresponds to \(A\) on the image is \(A'\), to \(B\) is \(B'\), and to \(D\) is \(D'\). We would read “\(A'\)” as “\(A\) prime.”

We can describe the translation using mapping notation as \((x,y)\rightarrow (x-2,y-1)\).

We are interested in the following:

• Compare the graph of a function \(y=f(x-h)+k\) to the graph of \(y=f(x)\). We want to develop a rule to describe the effect of \(h\) and \(k\).
• Sketch the graph of \(y=f(x-h)+k\), for specific values of \(h\) and \(k\), given the sketch of the function \(f(x)\), where the equation of \(f(x)\) is not given.
• Write the equation of a function whose graph is a translation of the function \(y=f(x)\).

For some function \(y=f(x-h)+k\), what is the effect of \(h\) and \(k\) on \(y=f(x)\)?

To investigate this use the following worksheet.

When you have finished the investigation, you should be able to describe the effect of \(h\) and \(k\) on \(y=f(x)\) when \(h\) is positive or negative, and when \(k\) is positive or negative.