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. |
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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. |
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