Assignment 3: Investigation of the quadratic equation
With an emphasis on b values
By: David Drew EMAT 6680, J. Wilson
††††††††† With graphing calculator it has become exceedingly efficient to show high school students different applications of the quadratic in respect to graphs. We now have the capability to graph ax2 + bx + c = 0 and overlay the graph of y = ax2 + bx + c for different value of a, b, and c as the other two are held constant.
††††††††† We start with the expression y = ax2 + bx + c where a and c are one and we use different values for b. In the previous exercise, Assignment 2, we looked at the different values of a when b and c were constants. Now we want to explore the equation deeper with a discussion of what happens if b and c are variables at different times.
††††††††† Weíll begin by letting a and c equal one. So our equation will be x2 + bx + 1 = 0. Therefore b will equal the equation
(-1- x2)/x = b. Hereís our first look at then equation. Do you have any conjectures and/or assumptions based on the knowledge youíve learned from Assignment 2?
††††††††† This new graph is very similar to our previous graphs but at the same time it is also much different. First of all there are two different parabolic shapes. In actuality these two curves are not parabolas. They are two different parts of what is called a hyperbola. But we donít want to discuss the hyperbola yet. Weíll just concentrate on the different intersections. Secondly they donít seem to intersect the y axis at all. Since we have substituted b for y we are now looking at the xb plane instead of the usual xy.
††††††††† Now if we throw in some random value for b then we get a line in the plane. Letís say we want to graph b = 4. What happens?
††††††††† We produced a line running parallel to the x axis. And notice that it intersects only one of our two curves. What does this mean? These two intersections are known as the roots of our equation. We can also guess from the picture that when b is equal to +/- 2 we will only get one intersection. Letís investigate this assumption. Weíll let b = +/- 2.
††††††††† It looks like our two curves hit the lines in only one place, namely +/- 2. So what does this mean and what happens if we have a value between -2 and 2? Hereís the last graph on this investigation. Weíll let b = 1.
††††††††† It turns out that this new line b = 1 does not interest either of our curves. So we say that when we have an equation like ours then if a line parallel to the x axis has a value of > 2 or < -2 we will have two intersections. But if the line is equal to the absolute value of 2 then weíll only have one exact intersection and hence one root. Lastly if the line is between 2 and -2 then we will have no intersections and no real roots.
††††††††† Letís shift our focus of our value of b and in turn look at the values of c. Since weíve exhausted the investigation of a in Assignment 2 and the values of b in the above discussion letís complete the exploration with c. Weíll keep the same equation as before and change the values of c, remember that c was one in our original equation. Letís start off by making c = 0.
††††††††† We get a straight line. Nothing too interesting here so letís suppose that c is a negative number. And if we remember correctly then we know that in the original equation c was positive, hence 0 = ax2 + bx + c. Letís see what happens when c = -1.
††††††††† This turns out to be extremely different from what weíve been working with for the last two exercises. Now we have what looks like a rotation of 900 about the z axis. Hereís two 3d views of our graphs so you can understand the rotation.
This is our new graph when c = a negative number compared to our old graph below.
Can you tell a difference and see the rotation?
††††††††† So in conclusion weíve seen several interesting features of our quadratic equation. If you have Graphing Calculator you can click on C Values to watch the cís change and how it affects the graph. But if you donít then hereís what happens. When c is less than 0 you will always have two roots for b. When c = 0 you will have exactly one root for b. And when c is greater than 0 you will have the same case as above when some places have two roots, two places has one root, and some places in the middle have no real roots.
Write Up By David Drew