by Donny Thurston

2. Fix the values for andb, varyc. Make at least 5 graphs on the same axes as you varya.aTry an animation for the same range.

What happens to (i.e., the case where b=1 and c=2)

as?ais variedIs there a common point to all graphs? What is it?

3. Fix the values for

anda, varyb. Make at least 5 graphs on the same axes as you varyc.cTry an animation for the same range.

What is happening mathematically?

Can you prove this is a translation and that the shape of the parabola does not change?

In this exploration, we will be looking in depth at the way that two constants

(aandc) influence graphs of the form:In looking at these graphs and exploring these ideas, we will be using still images that concurrently render several of the functions, as well as animations that help us see how the functions changes as a variable changes. See this exploration for discussion about the advantages that animations provide in math education and exploration.

First, lets examine what happens as the constant

avaries. Let us see an image with several values ofa, ranging from -3 to 4, and b = 1 and c = 1.Here we can clearly see that

ahas a tremendous impact on the graph, including changing what some might call the "shape" of the graph. For the most part, the "shape" of a parabola is generally described as the relationship between x and y, and basic transformations, such as translation, are not usually considered meaningful in changing a parabola's "shape". So what influence doesaactually have on the graph? I suggest thataactually determines therate of changebetween x and y, when we don't account for any translations of the graph that would be produced by adjustment incorb. Whilecorbmay adjust theintervalthat a particular rate of change may occur in, it isathat determines what that rate will be.We can observe this by taking the derivative of the function.

y = ax^2 + bx + c

y' = 2ax + b

In this instance the slope of the slope of the derivative is 2a. Recall that the derivative can be used to find the slope of the line tangent to the parent function at a given point by plugging in the x value for that point. For instance, if we plug in 1 for x, we discover that the slope of the tangent line is y = 2a + b. In our example above,

bwas always one, so we can discover that it is y = 2a + 1 whenever x = 1.This helps explain how the graphs change over varying

a.acontributes to determining the slope of the tangent line at any given x. To be exact, it works withbto make that determination, butbis a constant.adetermines the slope (which is specifically 2a) of the derivative.All of this tells us that

ais the rate of change of the rate of change of the function. This can be shown when taking the second derivativey'' = 2a

Since that is a constant, without any interference from

borc, we can see that at any given x,adetermines how quickly x is accelerating over time. If we conceptualizeaas a component ofacceleration, we can see how a largeacontributes to asteepgraph, and a smallacontributes to ashallowgraph. The "steepness" is the acceleration of x.Observe an animation of

achanging from 4 to -3.

What else can we observe from this animation? As per the question found at the beginning of this write-up, can we observe a common point? It is easier to notice the common point in the animation than it is the figure, I believe.

The answer is yes, there is a common point, in this graph it is (0,1). Why is this a common point in all of the graphs? It does not even appear to be the vertex in any of the parabolas, (and the straight line does not have a vertex).

It is a common point because it is the only point in which y (the output) is independent of the variable x (the input). As

achanges (andbdoes not), y changes accordingly, except for the instance when x is zero. If x is zero, thena(andb) can have no influence on the outcome of the function rule. Therefore, at the instance x = 0, y must be equal toc, being the only variable that is independent of x. As a result we can generalize that in any of the functions of the form : , asavaries, the common point between any of the graphs will always be (0,c).

So how does

cinfluence the function of the form: ?It appears to be a simple vertical translation that can also be seen here in this animation:

So what is happening here? Is the "shape" of the parabola really staying constant?

Well, mathematically, this is simply a vertical translation. This can be conceptualized as thinking of the graph "moving" up or down

cunits. How can we know this? By compartmentalizing the function rule.if y = f(x)

and f(x) = x^2 + x

then x^2 + x + c = f(x) + c = y + c

This shows that, for this specific case, adding any

cto the function rule maintains the behavior of that function rule, and simply translates y an equal amount. Or, in other words, for any input of x, y changesthe same amountas f(x) for anyc. This is becausecis not dependent on any value of x. Since "shape" is defined as the relationship, the behavior of the rate of change, between x and y, allcdoes is inflate ycunits, without respect to any x.

b,on the other hand, adjusts the function in a completely different manner from eitheraorc,and this is discussed in the next exploration found here.