EMAT6680 Assignment 2
Quadratic Equations
By Kevin Perry
Parabolas
Let us first explore the equation
If we select some values for the constants, for example A=1, B=4, and C=-3, the graph is
The first thing we want to explore is the effect of varying the different coefficients and observing the changes in the graph.
Changing the A Term
If we maintain the same values for B and C, and vary the A coefficient, we see the following family of graphs
One thing that we notice is that all the graphs pass through the point (0,-3), which follows from the constant C being equal to -3. We can see this by noting that no matter what A and B are, when x=0, the resulting y is always -3 (or C). We also see that as the absolute value of A gets larger, the parabola gets more tightly coupled and the slope of the endpoints gets larger and more extreme. Alternatively, as the absolute value of A gets smaller, the mouth of the parabola gets wider and the slope of the endpoints approaches a line. This line is a special line, and a special case; the case when A=0. When A=0, the equation reduces to
which is the equation for a line with a slope of B and a y-intercept of C. The absolute value of the slope of the sides of the parabola can never be any less than B.
So we can summarize the variations of A as when A gets larger, the parabola approaches a half-line with its end point at the y-intercept (the constant C). And as A gets smaller, the parabola approaches a line represented by y=Bx+C.
Changing the B Term
Next, we can fix A and C and vary B.
Let’s fix A and C so that
For various B, we get the following graph
We can see the basic shape of the parabola stays the same, but that the vertices of the different curves changes according to B. The vertex of the parabola goes more negative in the x and y direction as B gets larger and positive. But for B going negative and further negative, the vertex gets more positive in the x direction and more negative in the y direction. As B gets closer to zero from either direction, the vertex approaches the y axis at the common y intercept of all of the graphs. This point is again the value of C as above, and in this case is the point (0,-4).
The shape made by the vertices as B varies appears to create a parabola in itself. This shape can be derived and will be covered in a later discussion.
Changing the C Term
Finally, we must look at how the graph of the parabola changes if we vary C. As we have seen above, the value of C is an important point for the parabola, in that this is where the graph crosses the y axis.
For our discussion, we choose
The graph of these equations as C varies is
So we can see that the shape of the graph does not change at all, only that it is shifted up and down based on the value of C. And we now know that the value of C is the y intercept, and can conclude that as C gets larger, the graph moves up, and vice versa.
Conclusions
From this exercise, we have seen how varying each component of the general quadratic formula changes the graph of the parabola. We have seen that changing A affects the concavity and direction of the curve. With a changing B term, the graph changes its location with respect to the origin along a curve that may prove to be a parabola itself. And finally, by changing C we shift the graph up and down the y axis.
As an extension to this discussion, we can also observe how these changing graphs affect the roots of the equation Ax^2+Bx+C=0. We can see that the parabola either passes through the x axis twice for any particular graph, or it touches the axis at a single point, or it fails to pass through the x axis all together. We can conclude that for any quadratic equation, by looking at the graph of the function f(x)=Ax^2+Bx+C, whether there are real roots to the equation, and determine those roots by seeing where the graph crosses the x axis.
Try experimenting with a parabola.