Concept Byte: Graphing Inverses

For Use With Lesson 6-7

TECHNOLOGY

You can graph inverses of functions on a graphing calculator by using the DrawInv feature or by using parametric equations. It takes more keystrokes to set up parametric equations, but once you do you can easily change from one function to another and quickly see the graphs of the new function and its inverse.

Activity

Graph y = 0.3 x 2 + 1 y equals . 0.3 x squared . plus 1  and its inverse.

• Method 1 Use the DrawInv feature.

  • Step 1 Press y = begin box , y equals , end box  and enter the equation. Press zoom begin box , zoom , end box  5 to see a graph of the function with equal x- and y-intervals.

    
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  • Step 2 Press 2nd begin box , 2nd , end box   draw begin box , draw , end box  8. You will see DrawInv followed by a flashing cursor.

    Select equation Y 1 y sub 1  by pressing vars begin box , vars , end box   ► u 2  1 1. Press enter begin box , enter , end box  to see the graph of the function and its inverse.

• Method 2 Use parametric equations.
  • Step 1 Set to parametric mode. Press mode , begin box , mode , end box , comma  select Par, and press 2nd begin box , 2nd , end box   quit . begin box , quit , end box , .

  • Step 2 Enter the given equation in parametric form. Press y = begin box , y equals , end box  and enter the equations X 1 T = T cap x sub 1 cap t end sub , equals cap t  and Y 1 T = . 3 T 2 + 1 . cap y sub 1 cap t end sub , equals . . 3 cap t squared . plus 1 .

    

  • Step 3 Now use X 2 T = Y 1 T cap x sub 2 cap t end sub , equals , cap y sub 1 cap t end sub  and Y 2 T = X 1 T cap y sub 2 cap t end sub , equals , cap x sub 1 cap t end sub  to interchange the x- and y-values of the first parametric equation. Press y = begin box , y equals , end box  and move the cursor to follow X 2 T = . Select  Y 1 T cap x sub 2 cap t end sub , equals . , cap select . cap y sub 1 cap t end sub  by pressing vars begin box , vars , end box   ► u 2  2 2. Enter the equation Y 2 T = X 1 T cap y sub 2 cap t end sub , equals , cap x sub 1 cap t end sub  in a similar fashion.
  • Step 4 Press zoom begin box , zoom , end box  5. Adjust the Window so that Tmin and Tmax approximately agree with Xmin and Xmax. Press graph begin box , graph , end box  to see the graph of the function and its inverse.

Exercises

Graph each function and its inverse with a graphing calculator. Then sketch the graphs.

1. y = x 2 − 5 y equals , x squared , minus 5
2. y = ( x − 3 ) 2 y equals . open x minus 3 close squared
3. y = 0.01 x 4 y equals , 0.01 , x to the fourth
4. y = 0.5 x 3 − 3 y equals 0.5 , x cubed , minus 3
5. Writing Change the parametric equation X 2 T cap x sub 2 cap t end sub  = Y 1 T cap y sub 1 cap t end sub  in Method 2, Step 3 to X 2 T = − Y 1 T . cap x sub 2 cap t end sub , equals negative , cap y sub 1 cap t end sub , .  Describe the graph that results.
6. Explain how once you set up parametric equations, you can change from one function to another and quickly see the graphs of the new function and its inverse.

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