So if we convert our equation to something that can by graphed, we have:
With the use of a graphing calculator and some graphing packages, it
is easy for students to do their own manipulation at a faster pace. Being
able to produce multiple graphs quickly is important in order to keep the
attention of the students and to maximize productivity in the classroom.
Let's first examine the situation where a=1 and b=1. In this case, we will
vary c. We will use c=-2, -1, 0, 1, 2 and graphed the equations on the same
graph.
So, from this we notice that varying the value for c moves the parabola
up and down the y-axis.
Now let's examine the situation where b=1 and c=1. In this case, we will
vary a.
In the graph above, we see that all of the parabolas of this form go
through the point (0,1). Also we notice that the vertex of the parabola
changes as does its width. One may be interested in exactly how the width
of the parabola changes. It turns out that, if a < 1, the graph of the
parabola increases in width. The best way to understand this idea is to
look at the graph above. Compare the Red graph , where a=1 and the Cyan
graph, where a=1/4. As a increases the graph becomes more narrow and as
a decreases the graph becomes wider.
Now let's examine the situation where a=1 and c=1. In this case, we will
vary b.
In this case we see that the set of parabolas all go through the point
(0,1). Also it is apparent that the vertex of the parabolas shift as b changes.
When b is positive, the vertex shifts to the left of the origin and when
b is negative, the vertex shifts the parabolas to the right.
In all of the cases we have explored, the parabolas have been oriented so
that they opened up. Do all of the characteristics we have previously
discovered hold for a parabola oriented with the opening down?
Let's try a few parabolas with a negative a.
The parabolas change width the same as they do when a is positive. For
| a | > 1, the graph narrows in width and for | a | < 1, the graph
becomes wider.
So now let's put everything we've learned together. What do you predict
the graph of
will look like considering the things we've learned. It may help if you
compare the graph to the graph of the another parabola. Let's call it our
"standard" parabola with the equation:
From what we know, the graph should be wider than the standard parabola
since a = -1/4 and it should be opening downward, because a is negative.
Since c is positive the vertex of the graph will move up.
The rules for graphing parabols can be used by students to eliminate
the need for plotting many points. This increases the speed at which students
can produce graphs in the classroom which will in turn allow for exploration
into more significant matters or exploration of new ideas and ways of thinking.