## Articles

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 vertically c units.

• If c > 0, then the graph is translated upwards by c units.
• If c < 0, then the graph is translated downwards by c units.

Next we’ll look at what happens when we manipulate the a-coefficient, y = ax².

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 of a.

• If a < 0, the parabola is reflected in the x-axis (a frown).
• If a > 1 or a < –1, then the graph is stretched vertically (narrows).
• If –1 < a < 1 (and a ≠ 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 horizontally h units.

• If h > 0, then the graph is translated h units to the right.
• If h < 0, then the graph is translated h units to the left.