by Rives Poe
and Explorations with Second Degree Equations.
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.
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!
(You are on your honor!)
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.
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
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.
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?
Did you get it right? I
am sure you did. Now try when c= 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
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.
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
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
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
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.
Click here to investigate some different values of a on your
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