Game Development Reference
In-Depth Information
The graph you generate using a quadratic equation differs from the graph you
generate using a linear equation. The most fundamental characteristic of this
difference is that the slope of a parabola changes, whereas that of a straight line
defined by a linear equation does not.
To review this notion, consider the slopes shown in Figure 9.2. The slope of the
parabola tends to become steeper the larger the value of y becomes. On the other
hand, the slope of the linear equation remains constant throughout. The capacity
to show the change in the slope of a line provides you with a powerful tool with
which to examine rates of change. Central to this idea is that, as the value of the
x axis increases, you can discern a continuously steeper or more accentuated
change in the value of the y axis. With change comes a change in the rate of change.
As Figure 9.2 reveals, the change of the slope reveals a changing relationship
between the base number and the resulting value of y . Contrast what happens if
you add 2 to 2 or 3. Each time, you perform the same action. You add 2 to a
number. The slope of the graph that represents this activity stays the same. This
activity differs from what happens when you employ an exponent.
When you employ an exponent, if you raise 2 to the power of 2, the relation
between 2 and the base number changes as you increase the value of the expo-
nent. With each successive exponential operation, the ratios between the values
of x and y change, and the shape of the curve changes. When you can change the
slope in this way, you arrive at a way of representing or describing phenomena
that significantly expands the work you perform using linear equations.
Changing Appearances
The work you performed in Chapter 7 when examining absolute values anticipates
the work you perform with quadratic equations. As Figure 9.3 illustrates, you can
view the V shape of the graph you generate when working with absolute values as
similar to the rounded U shape you generate when you work with quadratic
equations. In both cases, the graph is symmetrical with respect to an axis. Two lines
meet to form the vertex of an angle. In the same way the point that corresponds to
the lowest (or highest) reach of the parabola is called the vertex of the parabola.
In most cases, the figures you generate are symmetrical with respect to the y axis,
but you can also generate graphs that are symmetrical with respect to the x axis.
Generally, however, if you apply the vertical line rule to the graph of a quadratic
equation, then functions you define using quadratic equations remain sym-
metrical with respect to the y axis.
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