Graphing Quadratics

When you graphed straight lines, you only needed two points to graph your line, though you generally plotted three or more points just to be on the safe side.

However, three points will almost certainly not be enough points for graphing a quadratic, at least not until you are very experienced.

For example, the most basic quadratic is y = x². Suppose a student, asked to graph this most-basic of quadratics, computes these three points:

Then, based only on his experience with linear graphs, he tries to put a straight line through the points.

He got the graph wrong. You, on the other hand, are more careful.

How do you graph a quadratic?

To graph a quadratic, start with a T-chart, plotting enough points that you can see the curvature of the graph. When you connect the plotted points, draw the curved line as curved, especially at the turning point (called the "vertex").

Suppose you are given y = x² to graph. You start by making a T-chart and finding many points:

That last point has a rather large y-value, so you decide that you won't bother drawing your graph large enough to plot it. (And this is perfectly okay! Not all computed points are equally useful for plotting. Use your good sense to pick and choose.)

Leaving aside that one point, you plot all the other points:

Even if you'd forgotten that quadratics graph as curvy parabolas, these plotted points will swiftly remind you of this fact.

So you draw a nicely smooth curving line, passing neatly through the plotted points:

Unlike the careless student, you just got the graph right.

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Some students will plot the points correctly, but will then connect the points with straight line segments, like this:

This is wrong! You do still need a ruler for doing your graphing, but only for drawing the axes, not for drawing the parabolas. Parabolas graph as smoothly curved lines, not as jointed segments. Take the time to draw it right.