thsalgebra Translations of Algebraic Functions
Translations of Algebraic Functions

In this lesson students will be using the graphing calculator to explore changes in algebraic functions. They will be given a function they have not yet learned and asked to determine how the placement of coefficients and constants affect the function. This will lead them to discover the following translations: horizontal shift, vertical shift, horizontal shrinking/stretching, and reflections. In addition, they will be asked to explain how some of these new translations are related to the slope of linear functions. By using the graphing calculator, the students will be able to focus on the concept and not have to worry about the mechanics of graphing the functions.

I believe a guided discovery of these translations is more effective than just a teacher demonstration. The students are able to understand the translations which helps them retain the different types of translations and be able to apply them to any function they encounter. This technique can be expanded later to use with trigonometric functions to show amplitude and phase shift.

Materials: graphing calculator, graph paper, pencil, colored pencils

Step 1: Graph the function y=x^2 on your graphing calculator. We will call this your primary function. Do not delete this from y1 in your “y=” screen. Sketch this function on your graph paper.

Step 2: Graph the following two functions as y2 and y3 respectively: y=x^2+3 and y=x^2–3. Using different colors, sketch these two functions on the same set of axes as your primary function. Under your sketch expain how the addition/subtraction of 3 affects the graph.

Step 3: Change your y2 and y3 to the following two functions: y=(x-4)^2and y=(x+4)^2. Using different colors, sketch these two functions on the same set of axes as your primary function. Under your sketch expain how the addition/subtraction of 4 inside the parentheses affects the graph.

Step 4: Again, change your y2 and y3 to the following two functions: y=2x^2 and y=0.5x^2. Using different colors, sketch these two functions on the same set of axes as your primary function. Under your sketch expain how a coefficient in front of the quadratic affects the graph.

Step 5: Finally, change your y2 to y=–x^2 and delete y3. Using a different color, sketch this function on the same set of axes as your primary function. Under your sketch expain how a negative in front of the quadratic affects the graph. Can you find a relationship with this and the slope of a linear function?

Now, given a general function y=a(x–b)^2+c explain how each letter affects the function. These changes are called translations. They are horizontal shift, vertical shift, horizontal shrinking/stretching, and reflections. Match each translation with its corresponding letter from the general function.
Last updated  2008/09/28 08:26:45 PDTHits  214