Graphing Quadratic Equations Using Transformations

Sometimes by looking at a quadratic function, you can see how it has been transformed from the simple function y = x 2 . Then you can graph the equation by transforming the "parent graph" accordingly. For example, for a positive number c , the graph of y = x 2 + c is same as graph y = x 2 shifted c units up. Similarly, the graph y = a x 2 stretches the graph vertically by a factor of a . (Negative values of a turn the parabola upside down.)

We can see some other transformations in the following examples.

Example 1:

Graph the function y = 2 x 2 − 5 .

If we start with y = x 2 and multiply the right side by 2 , it stretches the graph vertically by a factor of 2 .

Then if we subtract 5 from the right side of the equation, it shifts the graph down 5 units.

Example 2:

Graph the function y = − 1 2 ( x − 3 ) 2 + 2 .

If we start with y = x 2 and replace x with x − 3 , it has the effect of shifting the graph 3 units to the right.

Then if we multiply the right side by − 1 2 , it turns the parabola upside down and gives it a vertical compression (or "squish") by a factor of 2 .

Finally, if we add 2 to the right side, it shifts the graph 2 units up.

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