by Rives Poe

Assignment 2:

Problems and Explorations with Second Degree Equations.

INVESTIGATION 1

In this exploration, for the equation: y=x^2+x+c , we are going to experiment with the function, by changing values for c. The graph pictured below shows a parabola where c = 1.

y=x^2+x+c

Is there anything in this graph that could relate to c equaling 1? It's hard to tell now, but in just a few minutes, I bet you will know exactly what will happen with any value of c!

Here we go, let's try some more graphs on the same axis as we vary the third value. Try not to scroll down before you guess the graph will look like!

For the same equation: y=x^2+x+c, if we make c= -4 what would you expect the graph to look like? Will it be wider or narrower? WIll it move left or right on the x-axis? Will it move up or down on the y-axis? Is it possible it could be upside down? Try sketching a picture and then scroll down to see if you are correct.

y=x^2+x-4

How did you do? The graph looks very similar to the first, however it is lower on the y-axis and it looks a little bit wider. Where does it cross the y-axis? Can you tell? Let's see if you can do the next one.

Same equation: y=x^2+x+c. Let's make c= -2 this time. Remember, no peeking until you draw a sketch.

y=x^2+x-2

Are you seeing a pattern here? The parabola crosses the y-axis at y=-2. AHA, that is what our c value is!

Okay, now try for when c = 0. What do you think will happen? This time also try to figure out if the width of the parabola is changing.

y=x^2+x+0

What did you notice? The graph crosses the y-axis at y = 0. (I bet you all knew that - see I told you!) But what about the width of the parabola? Is it always the same? Let's investigate: if c=0, then we are left with y = x^2 +x. I am going to say x=1. Plug it in to the equation and you have y=(1)^2+1, which equals 2. So, when y = 2, x= 1. What if x =3? You would have, y = (3)^2 +3, so y= 12. It looks like for now, unless we put a coefficient infront of x, our parabola will always be the same width.

On to the next parabola:

What will our graph look like when c=2?

y=x^2+x+2

Did you get it right? I am sure you did. Now try when c= 4.

y=x^2+x+4

It seems that "c" determines the y-intercept. Let's look at all of the graphs combined, so we can be sure.

The graphs are all very similar, just located at different y-intercepts. Now, take some time and explore on your own. Try using decimals and fractions for the value of c, too!

Try some of your own values for c and observe how the graph changes as the value of c changes by clicking here.

INVESTIGATION 2

In this exploration, let's use the equation y=ax^2+x+c. We are going to fix the value of c, where c = 2 and change the value of a. The first graph below is where a = 1.

y=a*x^2+x+2

This graph looks very similar to the graphs in investigation 1. Think back to what we just learned about c in the previous investigation. It tells us where the y-intercept is. What do you think that a will tell us? Let's look at a = -9.

For a = -9

What do you think? It definitely looks different from the first graph. Not only is it skinnier, it is upside down! What do you think a graph will look like when a = -7? Sketch your idea and then scroll down to see if you are close!

For a = -7

It looks similar to -9, doesn't it? I think we need to try a fraction - let's try a = -4/5. Any ideas what will happen. I am pretty sure it will be upside down, but what about the width of the parabola. Scroll down to see.

For a =- 4/5

Well, the y-intercept is still y=2 and the graph is still upside down. But it definitely looks wider. Let's think back. When a = -9 we had a very skinny graph, when a = -7, it was still pretty narrow, but when a=-4/5 the graph is wider. If we compare -4/5 to the other numbers, we notice it is closer to zero, it is a larger number, I think we need to see more. Let's try a positive number, like a = 3/2.

For a = 3/2

Well, the graph is no longer upside down, the y-intercept is 2 and it looks kind of wide. The things we know for sure now are that when a is negative the graph will be upside down (as long as x is positive) and when a is positive the graph will look more like a horse-shoe. I think we need more examples to see how the width is affected by a. Next graph is y=4x^2+x+2.

For a = 4

Well, are you getting some ideas? It seems as though the farther away a is from zero the narrower the graph. Do you agree? Why is this , though? Let's give x a value for just a minute- if x=1, a=4, and c = 0 then our equation would look like: y=4(1^2) + 1+0 . Solve for y and we get y=5. That definitely makes a skinny graph! What if x=1 and a=1? Then our equation would be y=1(1^2)+1+0. Solve and y=2, this gives us a much wider graph. Compare these two below on the same coordinate plane.

y=4(1^2) + 1+0 (red graph)

y=1(1^2)+1+0 (blue graph)

Bring back our original equation: y=ax^2+x+2. Try and sketch the next graph before you scroll down. Let's say a = 7 here.

For a = 7

Our hypothesis seems to be right. The farther away from zero "a" is, the narrower the graph. Look at a = 9 and test this hyothesis. Does it hold true?

For a = 9

It looks like, we have figured that "a" value out!

In this investigation, it looks as though by vhanging the value of a, we can determine the concavity of the parabola, but also we can have an idea of it's width. Let's look at all of the graphs together. Do you think you can match the graphs with our explorations above? Give it a try and then check yourself by clicking on the link below.