# Explorations with e

Kasey Nored

This write-up is intended for high school students.

The first series of images are dealing with the following equation

Our first equation is

What do you think will happen to our graph above as we change a?

From our experiences with quadratic equations one might expect a dilation.

The red graph is , is this dilation as we know it? Or do you see this as more of contraction?

What about using a negative a? What would you expect? A reflection? About which axis?

Check out various a's graphed on the same plane.

What conclusions can you draw about changing a?

What do you see interesting about the y-intercept? Does that coincide with our a? What conjecture can you make from the y-intercept? Think about x's value where the graph crosses the y-aixs? What conjecture can you make about our equation? What conjecture can you make about the value of when x is 0? Does this remind you an exponent property? Did I really ask you all of those questions just so you can recall that any non-zero number to the zero power is one?

I don't know, you tell me.

Lets look at as c varies while a and b remain constant at one.

What would you expect based on previous experience? That c will cause a shift? A vertical or horizontal shift?

The purple graph is . No real surprises here, you should have expected a vertical shift from your earlier work with equations.

Let's move on to looking at changing b.

What do you expect? Recall our equation is we will set a = 1 and c = 0 as we vary b. Thoughts? Predictions?

The red graph is . What is happening? Is that what you expected? What about if b is negative?

Well now, what have we here? It appears that the negative b causes the reflection about the y-axis. Did you expect this? Should you have?

Let's investigate the image further. Looking at the black graph, it is obviously a reflection of . Looking at the curve what do you notice? What will happen if we use a fraction for b? Will the curve become steeper?

It appears that our curve dilates. Let's zoom out a bit.

While all of our graphs have the same y-intercept, it appears that it takes the red and blue graphs a bit longer to approach the asymptote.

Summation

What do you see? What connections can you make with the shifts and dilations of linear and quadratic equations? Are the rules the "same" or approximately so? Can you justify your conclusions? Could you write a list of "rules" for the shifts of ? What are those rules?