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In order to sketch the graph of a parabola given its standard equation, it is useful to find a few points which solve the equation, plot them on a coordinate plane, and then draw a parabola passing through these points.

In the following example we want to sketch the parabola y - 3 = (1/4)(x - 6)2.

First we must recognize that this equation is a transformation of the simple equation y = (1/4)x2. We know an equation of the form y = ax2 opens either up or down. And since in this equation a = +(1/4) > 0, we know this parabola opens up.

To practice plotting points to get the correct shape of a parabola, we start by graphing the equations y = x2 and y = (1/4)x2.

We want to form a table of values that will give us some coordinate points to plot on the plane. STEP 1: Choose six to ten values for x which surround the axis of symmetry. The first column in the table below contains seven integer values which surround the axis of symmetry, x = 0. STEP 2: Compute the equation y = x2 with these values of x. STEP 3: Plot all seven (x, y) coordinate points on a coordinate plane. On our graph, these points are marked by the blue dots. STEP 4: Draw a parabola with vertex at (0, 0) and axis of symmetry at x = 0 passing through all seven points. This is the red parabola on the graph.

Repeat the same steps to find points satisfying the equation y = (1/4)x2. We will use the same x values as before and add a third column onto our table.

This gives us some good information about how a effects the graph.

Notice that each point in column 3 is 1/4 that of the corresponding value in column 2. The effect on the graph can be seen by comparing the red and the green parabolas.

Since each y value on the green parabola is 1/4 that of the red parabola, we get a parabola that is much wider than the red parabola. The rule is the smaller the |a|, the wider the parabola.

x	y = x2	y = (1/4)x2
	0	0	0
	1	1	0.25
	-1	1	0.25
	2	4	1
	-2	4	1
	5	25	6.25
	-5	25	6.25

Our next step in graphing our desired equation y - 3 = (1/4)(x - 6)2 is to perform the translations. Recognize that the graph of this equation has exactly the same shape as the graph of y = (1/4)x2, only shifted up 3 units and right 6 units.

Knowing this, you could plot the parabola directly by redrawing the green parabola at a new position on the plane; or you can go through the steps of plotting points.

It is a bit redundant to go through the steps of plotting points here, since we already know what the graph looks like, but it is a nice exercise and a good way to verify the translation effects.

x	y = (1/4)(x - 6)2 + 3
	6	3
	7	3.25
	5	3.25
	8	4
	4	4
	11	9.25
	1	9.25