Write-up II
EMT 668
Some different ways to examine

by

James W. Wilson, Brian Seitz and Beth Richichi

It has now become a rather standard exercise, with available technology, to construct graphs to consider the equation

.


For example, let's give values to the a,b, and c variables and examine the graph of the following equation, where a=1, b=2, and c= -3:


Here, the lines are parallel to the y axis and intersect the x axis at x= -3 and x= 1.


The graph of a quadratic equation is a parabola.


Notice that the graph of opens upward, whereas the graph of
opens downward.

Now, let's examine the graph of

Notice that the y coordinate of each point on the graph is twice the y coordinate of the corresponding point on the graph of
.


Here, we have the same graph with a negative value of a:

Here, the y coordinate of each point on the graph is 1/3 the y coordinate of the corrsesponding point on the graph of .

If with a not equal to 0, then the graph has the same general shapes as that of or depending on whether a>0 or a<0. The graph, however, is more wide if |a|<1 and is more narrow if |a|>1. In either case the graph is symmetric with respect to the y axis. The origin is the maximum if a<0 and the minimum if a>0.
Let's examine the following quadratic equation:


The use of today's technology is of great help when examining correct solutions to the quadratic formula and the method of completing the square. When x=0, we see that y=-8, so the y intercept is -8. When y=0, the quadratic formula helps us to see that the x intercepts are x=2-(2/3)rt(3), x=2+(2/3)rt(3). Completing the square, we find that From this we know that the vertex of the parabola is (2,4). Thus we obtain the graph of the given equation by shifting the graph of upward 4 units and to the right 2 units. The use of today's technological methods, such as Algebra Xpresser, allows students to easily check their work.



We see varied values of a in the above graphs: a= 5, a=1/2, and a= -3. A negative value of a turns the parabola upside down with respect to the x-axis. As the value of a increases, the width of the parabola becomes more narrow. Similarly, a decrease in the value of a widens the parabola. The three parabolas meet at the point of locus, (0,4).



Here, the b value has been altered. A negative value of b shifts the parabola to the right of the y axis. Similarly, a positive value of b shifts the parabola to the left of the y-axis.

Alteration of the c value shifts the parabola up and down the y-axis. The width of the parabola remains the same. The parabola will intersect the y-axis at the value of c. For example, when c=-5, the parabola intersects the y-axis at (0,-5).


Since the three parabolas meet at (0,4), the locus is the parabola .


Now let's examine for different values of a, b, or c as the other two are held constant. For example, if we vary the value of b while a and c are held constant by setting , the result would be the following graph:


A positive value of b places the parabolas in quadrants II and IV. A negative value of b places each parabola in quadrants I and III. These graphs are inverses of each other and do not lie between y=-4 and y=4.


Setting values of b=-7, b=-4, b=3, and b=6 results in horizontal lines parallel to the x-axis. The lines b=-7 and b=6 intersect each parabola at two distinct points. The line b=-4 intersects each parabola at only one poiint, the maximum. The line b=3 does not intersect the parabola, since this line lies between y=-4 and y=4. From this we can determine that for b>4, the result will be two negative real roots. Similarly, for b<-4, the result will be two positive real roots. For b =-4 , the result will be one real root. For -4<b<4, there will be no real roots.

Similarly, the a and b values may be held constant while the c value varies. For example, let's examine the graph of :


Here, we have a graph in the xc plane, where a and b are held constant while c varies.



Notice that the c intercepts intersect the parabola at two points, except for the maximum, where the c intercept intersects the graph at one point.
The value of a can also be altered. Let's examine :



We see that the a-intercepts intersect the parabola at 0 or more points.

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