Graphing Quadratic Equations Using Transformations
A quadratic equation is a polynomial equation of degree . The standard form of a quadratic equation is
where and are all real numbers and .
If we replace with , then we get a quadratic function
whose graph will be a parabola .
Sometimes by looking at a quadratic function, you can see how it has been transformed from the simple function . Then you can graph the equation by transforming the "parent graph" accordingly. For example, for a positive number , the graph of is same as graph shifted units up. Similarly, the graph stretches the graph vertically by a factor of . (Negative values of turn the parabola upside down.)
We can see some other transformations in the following examples.
Example 1:
Graph the function .
If we start with and multiply the right side by , it stretches the graph vertically by a factor of .
Then if we subtract from the right side of the equation, it shifts the graph down units.
Example 2:
Graph the function .
If we start with and replace with , it has the effect of shifting the graph units to the right.
Then if we multiply the right side by , it turns the parabola upside down and gives it a vertical compression (or "squish") by a factor of .
Finally, if we add to the right side, it shifts the graph units up.