Game Development Reference
In-Depth Information
negative, the parabola opens downward. If the value of
h
is positive, then the
vertex of the parabola shifts to the right, or positive, direction on the
x
axis. If it is
negative, then the vertex of the parabola shifts to the left, or negative, direction
on the
x
axis. If the value of
k
is positive, then the vertex shifts upward on the
y
axis. If the value of
k
is negative, then the vertex shifts downward on the
y
axis.
The equation in this form proves so valuable that it is worthwhile knowing how
to convert quadratic equations so that they appear in this form. Toward this end,
consider again the standard form of a quadratic equation:
ax
2
þ bx þ c
To alter an equation you find in this form so that you can discern its component
variables, you perform an operation known as completing the square. The next
section covers in detail the procedure for completing the square of a quadratic
equation. For now it remains important to focus on the notion that the extended
form of the equation consists of a restatement of the standard form of the equa-
tion. Table 9.1 provides a summary of the features of the extended form of the
equation. Subsequent sections of this chapter discuss features not covered in the
previous sections.
Table 9.1
Features of the Standard Form
Item
Discussion
ax
2
þ bx þ c
Standard form of a quadratic equation. You can set the constants b and c to 0, resulting in
an expression of the form ax
2
, but you may not set the constant a to 0. By definition, the
quadratic equation is an equation of the second degree, so if you eliminate the term with
the coefficient corresponding to the second-degree variable, you change the equation
so that it no longer corresponds to the definition of a quadratic equation.
2
aðx hÞ
þ k
This is the form a quadratic equation can assume if you rewrite it by completing the square.
(h, k)
This coordinate pair establishes the position of the vertex. If h is negative, then the
vertex lies to the left of the y axis. If h is positive, then the vertex lies to the right of the y
axis. If the variable k is positive, then the vertex is above the x axis. If the variable k is
negative, then it lies below the x axis.
h
The value of h defines the line of symmetry for the parabola. If this is a positive value, then
the line of symmetry shifts to the right of the y axis. If the value is negative, then the line
of symmetry shifts to the left of the y axis.
a
The value of a determines how sharply the parabola rises. If the value is greater than 1,
then the parabola narrows and rises more precipitately. If the value is less than 1, then the
parabola becomes wider and rises less precipitately.
k
The constant k establishes the y intercept for the parabola. If the value of k is positive,
then the vertex moves upward relative to the x axis. If the value of k is negative, then the
vertex shifts downward.
