Translations of the Quadratic Function – An Exploration

By Sharon K. O’Kelley

 

This is an exploration for Algebra or Advanced Algebra teachers who want their students to explore how changes to a quadratic equation affect the graph of the parabola. This exploration focuses on the vertex form of the quadratic equation. (A key follows the end of the exploration.)

 

 1.  Consider the graph….

 

Let’s call this the graph of the “first function.”

 

 

The vertex is__________.

 

 2.  Now, let’s change the first function….

.

Compare the red graph with the original one. Describe the change and state the new vertex.

 

 3.  Now, ponder this change….

Compare the blue graph with the original one. Describe the change and state the new vertex.

 

 4. Do you want to see an animated version of what is going in questions 2 and 3? Go here. (Be patient. It takes awhile to load….)

 

 5. For the next part, we are going to hold “a” constant at 1 and vary “b.” Consider the following functions and their graphs.

    (black)

 (blue)

  (red)

Compare the blue and red graphs with the original one. Describe the changes and state the new vertices.

 

 6.  See an animation of these transformations. Click here.

 

 7. Sketch what you think the following graphs will look like…. (Be sure to note your vertices.)

       Go here to see the answer.

 8. Write the equation of the graph.

Equation:__________

 

 9. Compare the two given functions. Will they yield the same graph? Justify your answer.

10. Do these changes affect the shape or position of the first function?

 

11. Experiment on a graphing calculator or with graphing software and provide an example of a function which changes the shape of the parabola.

Function:__________________

12. Describe the transformations you have explored in this lesson. Connect what is going on in the equation with what you see in the graph.

 

13. Use what you have learned to write the equation for the red graph. Its original graph is given in black.

 

Original Equation:

Equation of red graph:___________

 

 

Key

 1. (0, 0)

 2. The parabola has translated to the right two units and its new vertex is (2, 0).

 3. The parabola has translated to the left three units and its new vertex is (-3, 0).

 5. The blue parabola has translated left 1 unit and up 2 units. The new vertex is (-1, 2). The red parabola has translated left 1 unit and down 3 units. The new vertex is (-1, -3).

 8.

 9. They do yield the same graph because the functions are equivalent. If 1 is subtracted from both sides in the first function, it becomes the second function.

10. These are translations which means they only change the position of the graph horizontally or vertically. The shape of the parabola does not change.

11. An example of a graph in which the shape of the parabola would change is

12. The graph of the parabola translates right a units when the function includes and left a units when the function includes . The graph translates up b units when the function has +b on the outside of the parentheses and down b units when it has –b. If b is on the same side of the equation as y, then the vertical translation follows the same pattern as the horizontal – e.g., +b would be down b units.

13.

 

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