Arguably, y = x^2 is the simplest of quadratic functions. In this exploration, we will examine how making changes to the equation affects the graph of the function.

We will begin by adding a coefficient to x^2. The movie clip below animates the graph of y = nx^2 as n changes between -10 and 10.

As you can see in the animation, the value of n effects the concavity of the graph and the rate of increase/decrease of the function.

Now, let us compare some specific values of n. We will begin by exploring how whether the coefficient n is positive or negative affects the graph of the function.

y = -3x^2	y= 3x^2

From these two graphs, what would you conjecture is the relationship between the value of the coefficient n and the concavity of the parabola?

I would then have students pick additional values of n and see if these values agree with the conjecture.

What happens when n = 0?

The rate of increase of a quadratic function describes how fast y is growing with respect to x.

Let us look at some examples algebraically and then graphically.

Consider y = 2x^2 versus y = 10x^2.

As x increases, the y-values in the y =10x^2 column get much larger than the y-values in the y=2x^2 column. For this reason, we say the rate of increase for y = 10x^2 is greater than the rate of increase for y = 2x^2.

Below, you can see these two functions graphed on the same axis. Which graph represents which function?

For a positive value of n, how would you conjecture the value of n affects the shape of the graph?

I would then have students pick additional values of n and see if these values agree with the conjecture.

In the previous, example we only looked at positive values of x. What will happen if we look at negative values of x?

As x gets smaller in value, y gets smaller in value. This means the function is decreasing. It may be more difficult to tell from looking just at the numbers that y=10x^2 is getting smaller more quickly than y = 2x^2. To compare them, we can formally look at the rate of decrease--meaning the change in y over the change in x.

Rate of decrease = change in y / change in x = (y2 - y1)/(x2 - x1)

We can pick any two points for each of the functions to compare as long as we pick the same x-values. For y = 2x^2, let us pick (-2, 18) and (-5, 50). Then for y = 10x^2, we will pick (-2, 40) and (-5, 250).

y = 2x^2: Rate of decrease = (50 - 18)/(-5 - -2) = 32/ -3 = -3.666666

y = 10x^2 :Rate of decrease = (250 - 40)/ (-5 - -2) = 210/-3 = -70

In general, we use the term rate of change, to describe how fast y is changing with respect to x. If as x gets larger in value, y gets larger in value, the function is increasing. If as x gets larger in value, y gets smaller, the function is decreasing.

Does a function have to either always be increasing or always be decreasing?

Given a function of the form y = ax^2 where a is positive, where is the function increasing and where is it decreasing?

Given a function of the form y = ax^2 where a is negative, where is the function increasing and where is it decreasing?

Previously, we have studied rates of change. In what context have we explored the change in y in relation to the change in x before?

Hopefully, the students recall that this is the slope of a linear function.

When we look at a non-linear functions, the rate of change of the function at a given point, is the slope of the line tangent to the graph at that point.

Below is the graph of y = x^2 and the line tangent to it at the point (1,1). Use the graph to calculate the rate of change of y =x^2 at the point (1,1).

Do you think the rate of change will be the same at every point on the parabola?

Below is an animation of the lines tangent to y=x^2 at each point. Does the animation below support or refute your claim?