Intermath | Workshop Support


Coeffecients that effect the graph 

Problem Statement
Explain how the graph of the function h(x) = ax2 + 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 = ax2 + bx + c   Then, if I let 2 = a and 0 = b and c, then my equation will read: y = 2x2  .  Our teacher brought her laptop with some really nice graphing software on it.  We graphed our equation, using different values for “a”:  y =  1 x2   was 1,1;  y =  2x2  was 1,2;   y =  3x2  was 1,3; y =  100x2  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 =  2x2 ; y =  2x2  + 3x;  y =  2x2  + 4x;  y =  2x2  - 3x;  y =  2x2  - 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

Link(s) to resources, references, lesson plans, and/or other materials
What is a function?
Link 2


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