Game Development Reference
In-Depth Information
This is not an issue if we are only concerned with the portion of the parabola
that is to the right of the vertex. In fact, by focusing only on that side of the
parabola, we can open up another bag of tricks with regard to the exponent.
By dispensing with the possibility of
x
values less than 0, we are allowed to raise
x
to fractional exponents (Figure 10.5). The result of doing this is the same as
increasing the exponent by whole numbers—the higher the exponent, the steeper
the resulting parabola. However, by allowing real numbers rather than integers, we
have much more flexibility in the exact shape of the curve.
FIGURE 10.5
The quadratic function
y = x
k
with the values of
k
as 1.5, 2.0, 2.2, and 2.5.
As before, as the exponent increases, the bowl of the parabola narrows.
Rotating the Curve
Another convenient manipulation is to use values of
k
that are between 0 and 1.
Remember that another way of writing the square root of
x
is
x
0.5
. Therefore, by
raising
x
to
k
values between 0 and 1, we are actually taking a root of
x.
The result-
ing curve is a parabola whose axis of symmetry is parallel to the
x-
axis rather than
the
y
-axis (Figure 10.6).
As with exponents greater than 1, we can change the shape of the parabola by
modifying the magnitude of the value of
k
. In this case, when we make the exponent
smaller, the parabola narrows. An easier way to remember this is for us to think in
terms of
closer to
or
farther away
from 1. Just as with
k
> 1, the farther away from 1
we move, the narrower the curve.
