by Donny Thurston
2. Fix the values for b and c, vary a. Make at least 5 graphs on the same axes as you vary a. Try an animation for the same range.
What happens to
(i.e., the case where b=1 and c=2) as a is varied?
Is there a common point to all graphs? What is it?
3. Fix the values for a and b, vary c. Make at least 5 graphs on the same axes as you vary c.
Try 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 (a and c) 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 a varies. Let us see an image with several values of a, ranging from -3 to 4, and b = 1 and c = 1.
Here we can clearly see that a has 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 does a actually have on the graph? I suggest that a actually determines the rate of change between x and y, when we don't account for any translations of the graph that would be produced by adjustment in c or b. While c or b may adjust the interval that a particular rate of change may occur in, it is a that 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, b was 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. a contributes to determining the slope of the tangent line at any given x. To be exact, it works with b to make that determination, but b is a constant. a determines the slope (which is specifically 2a) of the derivative.
All of this tells us that a is the rate of change of the rate of change of the function. This can be shown when taking the second derivative
y'' = 2a
Since that is a constant, without any interference from b or c, we can see that at any given x, a determines how quickly x is accelerating over time. If we conceptualize a as a component of acceleration, we can see how a large a contributes to a steep graph, and a small a contributes to a shallow graph. The "steepness" is the acceleration of x.
Observe an animation of a changing 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 a changes (and b does not), y changes accordingly, except for the instance when x is zero. If x is zero, then a (and b) can have no influence on the outcome of the function rule. Therefore, at the instance x = 0, y must be equal to c, being the only variable that is independent of x. As a result we can generalize that in any of the functions of the form :
, as a varies, the common point between any of the graphs will always be (0,c).
So how does c influence 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 c units. 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 c to 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 changes the same amount as f(x) for any c. This is because c is 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, all c does is inflate y c units, without respect to any x.
b, on the other hand, adjusts the function in a completely different manner from either a or c, and this is discussed in the next exploration found here.
