An Exploration of How the Value of the Coefficient a Effects the Graph of the Function y = ax^2 by Margaret Morgan (for College Algebra Students) 

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 decreasemeaning 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 xvalues. 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 nonlinear 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?
