For today's assignment, we will look at graphs of:
These graphs are parabolas that all open upward, with vertices at the origin (0,0). When a=1, the basic graph looks like this:
What will happen as we change the variable a? What will the graph do when we make 'a' larger than 1? What will happen when we make the 'a' some value between 0 and 1? What would happen if we make 'a' a negative number? The graphs of this equation quickly show us a pattern. So let's look at the graph of:
We can see that as ‘a’ goes from one to two, the parabola rises more steeply. Therefore we would expect that as a gets ever larger the parabola will narrow and rise even more steeply. So let’s view the graph of:
As we expected, the graph shows us a much steeper parabola. Now, what would happen if we make ‘a’ less than one? If making ‘a’ larger makes the graph rise more steeply, then we could expect that making ‘a’ smaller would make the parabola rise less steeply. So now we look at the graph of the equation:
Once again, the graph is as we expected. The parabola rises much more slowly. We have looked at positive values of ‘a’. Now let us consider what would happen if ‘a’ is negative? Let us review the pattern that has happened above. As positive ‘a’ gets larger the parabola rises more sharply. When we let ‘a’ be less than one, the parabola flattened out, approaching a straight line along the x-axis. So it would seem logical that if we continue to make ‘a’ even smaller, i.e. a negative number, it would start to curve the opposite way, opening downward. Let’s look at the graph of the equation:
Here, we let ‘a’ equal a negative one-tenth and it did open downward. It now appears obvious that as we let ‘a’ equal ever-larger negative numbers, it will open downward evermore steeply. Let’s view the graph of:
It confirms our idea that the larger negative ‘a’ would be a parabola that opens more sharply.
SUMMARY: The graphs of this function have a predictable pattern. This set of graphs would be good as a demonstration to a classroom of young math students. Putting up one graph, then questioning the students about expected outcomes as we change the value of ‘a’. After the second or third graph, the students would likely be able to correctly say what the outcome would be of the next change in ‘a’.