Stephanie Henderson's Assignment #2:

A Look at Quadratics

The standard
form of a quadratic equation is y=ax^{2}+bx+c, where a, b, and c are
rational numbers. Let's explore how each variable affects the graph of the equation.

To begin, let's look at a, when b and c are fixed at b=0 and c=0.

As |a| becomes larger, the parabola becomes wider, and as |a| becomes smaller, the parabola becomes more narrow. When a is positive, the parabola opens up and contains a minimum point, and when a is negative, the parabola opens down and contains a maximum point.

Now let's look and see if there's any difference when b=1 and c=1. Do you think a will have the same effect?

a continues to have the same effect as it did before, but changing our b and c values definitely had an impact as well. What point do you notice all graphs going through? Why do you think this occurred? Well, let's look at c's effect on the graph next. We'll set a=1 and b=1 and look at some different values of c.

According to these graphs, the parabola always has a y-intercept at the value of c. Let's try a different set of a and b values to see if this remains true. When we set a=-2 and b=2...

...the c value is still the y-intercept.

So now we've investigated how a and c affect our graphs. Lastly, we need to look at b.

When a=1 and c=0, our b value is the opposite of one of the x-intercepts. And in this case, our other x-intercept is 0, since we had a y-intercept of 0 in all of these parabolas. So let's look at another case where our y-intercept is something other than 0. What about a=-1 and c=2? Remember, our c value is the y-intercept, so these should all have a y-intercept of 2.

The b value is definitely not the x-intercept any more, so that is a special case when a=1 and c=0. However, there is a consistent relationship with our b value. If you look carefully, you'll notice that b is the horizontal distance from the y-intercept. In all of the parabolas above, a is negative, which creates parabolas that open downward. When this is the case, and b is negative, you move left b units from the y-intercept to get another point on the parabola. When b is positive, you move right b units from the y-intercept to get another point on the parabola. Now, when a is positive, and the parabola opens upward, the b value has an opposite effect. If b is negative, you move right b units from the y-intercpet to get another point on the parabola, and if b is positive, you move left b units from the y-intercept to get another point on the parabola. Try creating some of your own graphs to see these rules hold true.