Assignment 2 - Second Degree Equations

Faith Hoyt

Fix two of the values for a, b, or c. Make at least 5 graphs on the same axis as you vary the third value.

Our general form of a second degree equation is . Now, we want to investigate three different graphs, where we fix two of the variables and vary a different third variable.

First, let's look at the general form of this equation, so we have something to compare to for the rest of our graphs. If we simply plug in our above equation to graphing calculator, this would be the result:

For our first graph, we want to fix a = 1 and b = 2. We are going to vary our c values. I chose to use c = -10, -5, -2, -1, 0, 1, 2, 5, and 10. Below you can see the graph where each equation is graphed on the same axis:

When we compare this to our original graph above, notice that the only movement this graph does is a vertical shift. The y intercept is whatever we set c equal to. Thus, this causes the graph to move up or down, while the general shape of the parabola remains the same. Therefore, we can predict that by changing the value of c, we are simply moving the graph vertically by changing the y-intercept. Here, you can see an animation of the graph as we change c. In this animation, you can again note that the shape of the parabola does not change.

Now, let's investigate a different graph where we fix two different variables and vary a different third variable. In this case, for our second graph we will fix b and c and vary a. Let's fix b=1 and c=3. If we don't use anything for a yet, we notice the differences in this graph as compared to our first graph. Our graph still has the similar shape, but is simply moved up as we fixed our c value at 3. Now, we want to see what affect changing a will have on the graph. For this study, we will let a = -5, -3, -1, 0, 1, 3, and 5. Below, you can find the graph where each equation is graphed on the same axis.

This graph is much more interesting than our previous investigation. We notice that a has a much bigger impact on the graph itself than c does. In fact, a determines the direction of the graph. For example, when a=-5, the graph is concave down and when a = 5 the graph is concave up. Generalizing this, when a is negative, the graph will open downwards and when a is positive the graph will open upwards. Further more, a also changes the shape of our parabola. First, notice when a is 0, the equation is no longer a second degree equation, but rather a linear equation, giving us the line with the y intercept at y=3 and the x intercept at x=-3. For our other values, first let's look at when a is a negative. We can observe that as our a value gets closer to zero it is getting larger, or wider. As it moves away from zero, the graph beings to get skinny. Similarly, for our positive a values, the same is true. The closer the value to zero, the fatter the graph will be. As we begin to move away from zero, our graph begins to get skinny, or go on a diet. Thus, a affects the direction of the graph as well as the stretch of the graph.

We can see this even further if we look at the animation where we can observe even more values for a (in this case from -10 to 10).

Now, let's look at one last graph to investigate the effect our third variable will have on the graph. In this graph we will be fixing a and c, while varying our variable b. So, let's set our a=1 and c = 2. First, let's see what we would get compared to our first graph, if we don't substitute anything in for b yet. By just looking at the graph, we notice that b seems to have little affect on the graph. If anything it affects the stretch. So, let's substitute some values in for b to see what the affect is. For the purpose of this investigation, I will set b = -6, -4, -2, 0, 2, 4, and 6. Below you can find the graph where each equation is graphed on the same axis.

Now, seeing all the different graphs put together on one axis, we notice that b does in fact have an impact on our graph, and a rather large on at that. By looking at our graphs, we can see that some are reflected over the y axis, or at least seem to be. For example, let's compare the black and purple graphs. Notice that in the purple graph, b=-6 while in the black graph, b=6. This variable is known as the linear variable, which we can confirm as it is aligned with the x term. So, notice that when our b values are negative (as in our purple graph) our graphs tend to be on the positive x axis and when our b values are positive (as in our black graph) our graphs tend to be on the negative x axis. They all still intersect the y axis at 2 as that is our c value. Thus, we can generalize that our b value causes the graph to shift left and right, as well as up and down. We can see this more so in the following animation, where I set the boundaries for b from -10 to 10.