Coeffecients
that effect the graph
Problem Statement
Explain how the graph of
the function h(x) = ax^{2} + bx + c
changes when you modify a, b, and c.
Plans to
Solve/Investigate the Problem
To solve this problem, I will probably have to guess
and check with one variable at a time, and see what happens to the
parabola, using a graphing calculator.
Investigation/Exploration
of the Problem
First let me say that without the able help of our
teacher and my math wizard classmates, I don’t think I could have
ever done this problem, by myself.
That said, the first thing I had to
do was to find out what a “function” was. Intermath
dictionary says that a function is
In other words, for every value I put in for an
independent variable, I will only get one value in my range (or dependent
variable).
Next, my class taught me that h(x) can also be written
as “y”, in this case. Therefore, my equation can now
read: y = ax^{2 }+ bx + c Then,
if I let 2 = a and 0 = b and c, then my equation will read: y = 2x^{2
}. Our teacher brought her laptop
with some really nice graphing software on it. We graphed our
equation, using different values for “a”: y = 1 x^{2}
was 1,1; y = 2x^{2} was 1,2; y
= 3x^{2} was 1,3; y = 100x^{2} was
1,100. We found that as we increased the absolute value of
“a”, the parabola got skinnier. Also, as the absolute
value of “a” was increased, “y” got bigger, too.
Then we tried making the value of “a”
a negative number, and found that the parabola turned upside down, but the
absolute value of “a” still corresponded to
“y”.
The second variable we worked on was the
“c”. We quickly found out that “c”
represented where the parabola would cross the y axis, in the equation
given. By changing only the “c”, the parabola just slides
up and down the y axis, not changing its shape.
The last variable we worked on was the
“b”. I didn’t write down what the x,y
coordinates were, and I don’t have a graphing calculator to duplicate
what we did, but our class graphed the following: y = 2x^{2}
; y = 2x^{2} + 3x; y = 2x^{2}
+ 4x; y = 2x^{2}  3x; y = 2x^{2}
 4x. When we plugged these equations into the graphing calculator,
it made the vertex of the parabola change from side to side, and up and
down.
In summary, changing the “a” variable
determined the width of the parabola; changing the “b” variable
determined where the vertex of the parabola was; changing the
“c” variable determined where the parabola would cross the y
axis.
I learned about “translation”,
“reflection” and “rotation”, as we played with the
variables. Translation makes the parabola slide up and down.
Reflection flips it/inverts it. Rotation changes it from side to
side. It was helpful to see examples of what we talked about.
Meg Ramsey
mramsey@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other
materials
What is a function?
Link 2
U
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