**Transformation of Parabolas**

by

**Jeffrey R. Frye**

This discussion and illustration will show how changing the equation will change the display of the graph. The basic equation will be adjusted by a constant that will change the position of the graph relative to the x-axis and then a different constant that will change the position relative to the y-axis. The graph will then be reflected about the original vertex. The original equation is:

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When the graph is adjusted by substituting (x-4) for x, the graph is translated 4 units to the right. This is the red graph.

In order to move the graph to the 2^{nd} quadrant
the value (x+4) is substituted for x and a constant of 4 replaces the -4. The +4 translates the graph up 4 units and
the (x+4) translates the original graph 4 units to the left. This is the blue graph.

After changing the equation to vertex form, it is easier to see that changing the sign of the coefficient preceding the x will reflect the graph about the original vertex. The original equation in vertex form is shown below. Changing the sign to negative (-2) results in the reflection. This is the green graph.

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The graphs show the effects of the above transformations.

To explore these transformations using Graphing Calculator,
click here.

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