In unit 2 you learned
that by making certain adjustments to the rule of a quadratic
function of the form , you could
shift the graph up or down or side to side. Hopefully you also
found that the ordered pair (h,k) was actually the vertex of the
parabola. To finish our exploration of transformations of quadratic
functions, we will take a look at what happens when we add a coefficient
to the function. Namely, we will look at the function of the form
** **and try to determine what
happens to the parent function when we change the value of *a*.

Graph the parent function on your graphing calculator and then compare it to each of the following functions. Briefly describe the transformation of the graph of as compared to each function below.

Use your own words to describe the similarities and differences of these graphs.

Is this what you observed?

Now graph the following graphs and compare them to the corresponding graphs above.

Is this what you observed?

Write a few sentences describing what happens when the coefficient of is greater than one, between zero and one, between negative one and 0, and less than negative one.

Now graph each of the following pairs of graphs:

Use the language that we have learned (shift, reflection, stretch/compress) to describe how each of these pairs of graphs relate to the parent function. Then write another sentence for each pair describing similarities and differences within each pair.

Now we can put all of our work together...

Without the aid of a graphing utility, sketch a graph of each of the following functions and describe all of the transformations of the parent function that lead to your final graph.

Take another look at the function g(x) above. Does it seem familiar? Look back at Unit 1 and compare the graph of this function with the graph of the projectile data. Our goal in Unit 4 will be to try to find a rule of the form that best fits some given data that appears to be parabolic in nature.

Move on to Unit 4 - Fitting Functions to Parabolic Data.