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Exploring the Vertex Form of the Quadratic Function

Equipment needs:
• TI-83+ calculator with Interactive Graphing installed .

Studying the effect of A, B, and C
1. Press [APPS] and select the Transformation Graphing by
pressing the number to the left of Interact. Press any key
(except y or ) to install Interactive Graphing.
(Note: If you do not see the screen illustrated to the far right,
select 2:Continue)
2. In Func mode, press   to display the y= editor. Clear any functions that are
listed and turn-off any plots . Enter the general vertex form , Y = A(X-B)2+ C of
the quadratic function . Press
C.
If the Play-Pause Mode is not selected to the left of Y1 (>||), press until the
cursor is over the symbol and then press until the correct symbol is selected.

NOTE: You entered Y=A(X-B)2 + C in place of Y=A(X-H)2 + K, which is the form commonly found
in textbooks, because Interactive Graphing only uses the coefficients A ,B, C, and D.
3. Press  to display the settings screen for Transformation Graphing.

As a starting place, set the SETTINGS as pictured. To make these selections, press This defines the starting values for the coefficients and the increment by which you want to observe the change in the coefficients .
Studying the Effect of B
4. Press 6:Zstandard to display the graph. The graph will show the pre-
selected values of A, B, and C. Both the X and Y-axis range from -10 to 10 with
a scale of 1.
Press to cursor down one space and highlight B=. You will start your study
with the effect of B.

5. Press to increase the value of B by the pre-selected Step value (1 in this example). The graph is
automatically redrawn showing the effect of this change on B. Continue to press until you have an
idea of how changing B effects the graph.

6. Press   to decrease the value of B by the pre-selected Step value. Did the graph move the direction
you would have expected?

** Changing the value of B has the curve move in what direction?

This moving of the curve is called a translation in the X-direction or horizontal translation.

Use the cursor keys ( to change the value of B again. As you change B notice the x-coordinate of the
vertex.

** If B=3 where is the vertex? How about B=5? And when B=-2?

Make a hypothesis about the relationship between B and the vertex of the parabola . Test your hypothesis
by entering B=1, B=3, B=5, B=-1, and B=-2. Were you correct?

In vertex form Y = A(X - B)2 + C, the value of B gives the x-coordinate of the vertex. Be careful, though.
Notice the form has X-B. If you had the equation Y =(X-3)2 then B=3 and the vertex is at x = 3. For the
equation Y=(X+1)2 the B would be -1 with the vertex at x = -1.

Studying the Effect of C
1. Press   to highlight the C=. Press the   several times and notice the change in
the graph. Press several times and notice this change.
2. Predict where the vertex of the function will be if you let C=2. Enter 2 for C and
check you prediction.
3. Make some conclusions about the effect of changes in C on the vertex. Check your conclusions by
entering test values for C.
Studying The Effect of A
1. Return to the Transformation Graphing Screen and press   until the A= is
highlighted.
2. Use the same discovery method you used with B and C to investigate the effect of A on the graph of
the parabola. Be sure to let A be both negative and positive . Continue with the next question once you
have a hypothesis, and have checked it, for the effect of A

** What effect does changing the value of A have on the graph? Be sure to discuss both magnitude and
sign changes .
Quick and short check…..
** What is the equation of the parabola (quadratic function) graphed to the
right? (Note: The scale is 1)
** Graph the equation y = 3(x +1)2 -5 by hand. Check your answer with your calculator.
You might want to deactivate Interactive Graphing before continuing.
1. Press [APPS], select the number preceding Interact
2. Select 1:Uninstall
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