Observation 1:

Did you notice that for values of the coefficient that were greater than 1, the graph seemed to get thinner and for values between 0 and 1, the graph seemed to get fatter. This gives us some idea of the shape of the graph. However, what is really happening here is the graph is stretching vertically for values of the coefficient greater than 1 and it is compressing vertically for values of the coefficient between 0 and 1. This is easier to understand if we look at the data comparing one of these graphs to the parent function. The chart below should be a comfortable example of this transformation using the parent function and the data for the rule .
x  f(x)=x^2  f(x)=3x^2 
5  25  75 
4  16  48 
3  9  27 
2  4  12 
1  1  3 
0  0  0 
1  1  3 
2  4  12 
3  9  27 
4  16  48 
5  25  75 
Notice that for each value of x, the y value is three times as large for the function as it is for the parent function. Compare the y values of the parent function with those of the function . Do you find that the y values are five times smaller in the latter function (or one fifth as large) for every x value?
Did you find that each graph was reflected or flipped over the xaxis so that each graph now had a vertex that was the maximum point on the graph? In fact, as you will discover in the final investigation of this unit, the graph is actually reflected over its vertex.