Game Development Reference
In-Depth Information
situation. On the left, you see a parabola defined with a coefficient of 1. In the
middle, the value of the coefficient becomes
1
2
. On the right, the value of
the coefficient becomes
4
. As the value of the coefficient decreases, the slope of the
line tends to become more gradual.
Narrowing a Parabola
As you might expect after experimenting with values less than 1 in relation to the
coefficient of
x
in a quadratic equation, when you make the coefficient of
x
greater than 1, you increase the steepness of the parabola's climb (or slope).
Figure 9.6 illustrates how this happens. On the left, the parabola you see is
defined by an equation in which the coefficient of
x
is 1. In the center, the value of
the coefficient increases to 2. The steepness of the climb increases. On the right,
the value of the coefficient increases to 3. The steepness of the climb becomes
even greater. In each case, as the value of the coefficient increases, the steepness of
the parabola becomes more pronounced.
Translation Along the x and y Axes
When you graph a quadratic equation, in many instances, you find that the
vertex of the parabola you generate corresponds to the origin of the Cartesian
plane (0, 0). The discussions of linear equations in Chapter 7 and elsewhere
emphasize that while many linear equations cross or intersect with the origin of
the Cartesian plane, by shifting an intercept value, you can raise or lower the
intercept point. The same situation arises when you work with quadratic
equations.
When you consider the coordinate pair that defines the vertex of a parabola, you
deal with the minimum value of the parabola if it opens upward. You deal with
the maximum value of the parabola if it opens downward. You can shift this
value so that it moves along the
x
axis or along the
y
axis. If you move it along the
x
axis, you translate it horizontally. If you move it along the
y
axis, you translate it
vertically. You translate the vertex in a positive fashion if you shift it up or to the
right. You translate the vertex in a negative fashion if you translate it downward
or to the left.
To understand the mechanics of translating the vertex of a parabola along the
x
axis, consider an expression of the following form:
ax
2
