The window must be very small to show these 5 functions clearly.  The black function is the original function y = ax².   Notice that as the value of a increases, the width of the parabola decreases.  This pattern is consistent with both rational and irrational numbers greater than one. 


Next, take these same equations and make each value of "a" negative. 


                          

                              

This time the window is slightly larger.  This enables us to get a larger scale view of what is happening to these graphs.  Notice that in each case, when "a" becomes negative, the graph of the function is reflected over the x axis.   Remember that we're still restricting our investigation to values of "a" whose absolute value is greater than 1.

We'll now explore values of "a" greater than -1 and less than 1.

              


Notice first that these equations are again reflected over the x axis when "a" is negative.  The black function is our original function y = x².  When 0 < a < 1,  the function is wider than the original function.  As the absolute value of "a" becomes smaller the function becomes wider.

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We can summarize our findings as follows:

Any equation of the form y = ax² will always be centered at the origin. 

When a > 0,  y = ax² opens up.
When a < 0, y = ax² opens down.
When a = 0, y = ax² is a horizontal line at y = 0.

When |a| > 1, y = ax² is narrower than y = x².  The larger |a|, the more narrow y = ax².
When  |a| < 1, y = ax² is wider than y = x².  The smaller |a|, the wider y = ax².

In general, |a| and the width of y = ax² have a direct negative relationship.

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Now that we have explored the function y = ax², try the following on your own.

Tell in words what the graphs of the following equations will look like compared to the graph of y = x².

a) y = (-2/3)x²
b) y = (-3/2)x²

Which of these graphs will be wider, a or b?

c) y = 5x²
d) y = 2.5x²

Which of these graphs will be wider, c or d?


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