The simplest quadratic equation is:

y = x²

If you were to graph this using a table of values, it would look like the graph on the left.

Notice that vertex is directly at **(0, 0)**. But a world where all parabolas are fixated at the **origin** would be boring. In this lesson, we will learn how to move quadratic around the x-y plane algebraically. This is known as **transforming **a quadratic equation.

Let’s see what happens with we add or subtract a constant, c, to y = x² **± c**.

- Note that some textbooks use “k” to denote the constant!

The take-home message is:To graph y = x²

± c, translate the graph of y = x² verticallycunits.

- If c > 0, then the graph is translated upwards by
cunits.- If c < 0, then the graph is translated downwards by
cunits.

Next we’ll look at what happens when we manipulate the **a**-coefficient, y = **a**x².

The first video will show you how a +*positive* **a**-coefficient affects the **stretching/compressing** of the parabola. Two example are provided, so be sure watch both examples!

If you’d like to reflect the quadratic, so that it’s facing down, make the a-coefficient –*negative*.

The take-home message is:To graph y =

ax², stretch or compress the graph of y = x² vertically by a factor ofa.

- If
a< 0, the parabola is reflected in the x-axis(a frown).- If
a> 1 ora< –1, then the graph is stretched vertically(narrows).- If –1 <
a< 1 (anda≠ 0), then the graph is compressed vertically(widens).

Finally, in order to translate the parabola **left** and **right** from the origin, a constant, **h**, must be added or subtracted to the initial x-term, like this: y = (x **± h**)².

The take-home message is:To graph y = (x

± h)², translate the graph of y = x² horizontallyhunits.

- If
h> 0, then the graph is translatedhunits to the right.- If
h< 0, then the graph is translatedhunits to the left.