Parabolas
The Graph of .
Figure 1. A parabola
The
graph of an equation involving and is the set of all
points in the
cartesian coordinate plane
whose coordinates satisfy the equation.
The
graph of a function is the graph of the equation
For example, Figure 1 shows the graph of the equation
This particular graph is an example of a parabola. Note that it
is symmetric with respect to the -axis which is called its axis or line
of symmetry. The lowest point on the parabola (which in this case is the origin)
is called its vertex.
We'll use this graph and equation to illustrate some
ideas that have much wider applicability. Let's consider making some
changes:
Rescaling
Suppose we multiply the value of with a constant. Let's call it
.
Thus we obtain the new equation
If this is the same equation as before. Let's consider some other
possibilities, however:
Figure 2. Several parabolas
Figure 2 shows the graphs of these equations for
Graphs can be similarly reflected, stretched or compressed in the
horizontal direction by multiplying with a constant. However,
in the present simple example this effect is equivalent to a
vertical rescaling since
where .
Translations
Consider now the effect of subtracting a constant from :
This is equivalent to
and so it has the effect of
raising the graph of by units. Of course, if is
negative, the graph is lowered (by units).
Subtracting a constant from has the same effect in the
horizontal direction. Consider the equation
and
compare it with the original equation . As in ,
is never negative. It assumes its minimum value () at .
Thus the graph of is that of shifted units to
the right. If is negative, the graph is shifted units to
the left.
Figure 3. Translated Parabolas
Horizontal and vertical shifts can be combined, and any such
combination is called a translation. The general form of a
translation applied to is given by
which is
often written as
A translation does not change the
shape or orientation of a graph. It only changes its
location. Figure 3 shows translations of the graph of
with
The Completed Square Form
The completed square form of a quadratic function is
Its graph is a parabola whose vertex is .
If it opens up, if it opens down. (If the
graph is a horizontal line which you can think of a s a degenerate
parabola.) The vertical line is the axis or line of
symmetry of the parabola.
It is clear how a general quadratic polynomial
can be converted to the standard form. Just complete the square, as
illustrated in the following example:
The graph of this particular equation is given in Figure 4.
The resulting parabola has and the vertex (. It is
obtained from the standard parabola by reflecting it thorough
the -axis, stretching it vertically and translating it 1 unit to
the left and 8 units up. Note that all of these properties can be
immediately
read off the equation
Figure 4. Another Parabola
Scaling and Translating Graphs
The principles illustrated here apply to any equation, so let's
restate them:
- A combination of horizontal and vertical shifts is a
translation of the graph, a combination of horizontal and
vertical compression and stretching is a scaling of the
graph.
- Adding a constant to shifts the graph units
to the right if is positive, and to the left if is
negative.
- Adding a constant to shifts the graph units up
if is positive, and down if is negative.
- Multiplying with a positive constant compresses the
graph horizontally if and stretches it horizontally if .
If is negative these effects are combined with a reflection
through the axis.
- Multiplying with a positive constant compresses the
graph vertically if and stretches it vertically if .
If is negative these effects are combined with a reflection
through the axis.
As usual, I recommend that you do not memorize these facts. Rather
think about what's happening in terms of the interplay between the
graph and the equation, and figure out the effect of what you are
doing, or what you need to do, rather than relying on remembering
arcane facts.