Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Translating and Scaling Graphs

If you see this red text then unfortunately your browser does not support applets. You may have to enable Java, or use a different system.Click here to see a gif image of the applet that you should be seeing.

The applet just above this paragraph on this page can help you understand the effects of simple shifts (or translations) and rescalings of a graph. You can either look at the graph and deduce the equation, or start with the equation and figure out the graph. There is a detailed explanation in terms of parabolas. Here we give a summary in terms of a general function $ f $, and then explain the operation of the applet.

Suppose we start with a basic function $ f $ and consider its graph, i.e., the graph of the equation

$\displaystyle y=f(x). $

This is the basic graph and you can have the applet display it for the following special choices of $ f $. Click on the description of the function to see its graph. (Use the Back button of your browser to return here.)

Consider now the function

$\displaystyle g(x) = af(x-h) + k $

for some constants $ a $, $ h $, and $ k $. Naturally, the graph of $ g $ will be similar to that of $ f $. Specifically, the graph of $ g $ is obtained from that of $ f $ by
  1. Shifting the graph of $ f $ $ h $ units to the right. This is a horizontal translation of the graph.
  2. Multiplying the $ y $ value of the shifted graph by $ a $. This is a scaling (or rescaling), of the graph.
  3. Translating the shifted and scaled graph up by $ k $ units. This is a vertical translation of the graph.

These ideas are illustrated for

$\displaystyle f(x) = x^2,\quad h=2, \quad k=-3, \quad\hbox{and}\quad =2 $

in the nearby Figure.

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Figure 1: An Example.

The blue graph is the graph of $ f $. The red graph is the graph of

$\displaystyle g(x) = 2(x-2)^2 -3 $

corresponding to $ a=2 $, $ h=2 $, and $ k = -3 $.

This Figure was created with the applet on this page. The remainder of the page describes how to operate the applet.

Operating the applet.

In what follows we'll refer to the values of $ h $, $ k $ and $ a $ as the data and to the graph of the corresponding function simply as the graph or the modified graph.

To use the software, follow these steps: