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Graphing Quadratic Equations Using Transformations

A quadratic equation is a polynomial equation of degree $2$ . The standard form of a quadratic equation is

$0 = a x^{2} + b x + c$

where $a, b$ and $c$ are all real numbers and $a \neq 0$ .

If we replace $0$ with $y$ , then we get a quadratic function

$y = a x^{2} + b x + c$

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 $y = x^{2}$ . 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 $2$ .

Math diagram

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

Math diagram

Example 2:

Graph the function $y = - \frac{1}{2} {(x - 3)}^{2} + 2$ .

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.

Math diagram

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

Math diagram

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

Math diagram