ASSIGNMENT 1:

An investigation of the exponential function

In this assignment, we will be investigation the exponential function y=e^x. First, we will look at the general form, y=ae^(bx)+c.

In the graph to the right, you will see this general form represented by the red curve with a,b,c=1. You can see it is very similar

to the purple curve of y=e^x. It is moved up one on the y-axis. This makes sense because with a,b,c=1, the expression becomes

y=e^x+1.

Building on letting our c value equal a nonzero number, let's see what happens when it equals 5.

Again, we notice the curve is move up, this time by 5.

Here we let c=-5. As expected, the curve moves down by 5.

Now, let's see what happens when we let a,b=1 and c vary between -4 and 4. Notice that

when c increases, the curve becomes more stretched out and the y values do not increase as

rapidly. When c decreases, there is a more pronounce bend in the curve as the exponential

growth becomes much more drastic.

Lets investigate the a term now.

A series of graph are shown below with a=30, a=5, a=-5, a=-30. Both b,c=1.

a=30_________________________a=5

a=-5_________________________a=-30

Now let's see an animation with b,c=1 and a varying -10 to 10.

As you might expect, when a=0, the curve becomes a straight horizontal line passing through y=1. Why? Letting

a=0 takes the exponential component out of the equation leaving y=c=1.

You will also notice that as a changes from postitive to negative, the graph is reflected across a horizontal line, this

line equalling the c value.

In the above graphs, you will notice that as a gets further away from zero, the curve has a more drastic bend.

This means the curve is increasing more rapidly when a is positive and going away from zero. It is

decreasing more rapidly when a is negative and going away from zero.

Ok, now it's the b value's turn.

Below are a series of graphs with b=10, b=2, b=-2, b=-10. Both a,c=1.

b=10________________________________b=2

b=-2_______________________________b=-10

And the animation for a,c=1 and b varying between -10 and 10.

You can there is a horizontal asymptote at y=1 which is the c value. You can also see the curve

forms a horizontal straight line at y=2 when b=0. This makes sense because e^0=1. So the expression becomes

y=ae^0+c. And with a,c=1 then y=1+1=2.

You can also see the graph is reflected across the y-axis as b changes from positive to negative. From the graphs

above you can see that as b moves away from zero, the curve again bends more drastically. Again, this means that

the curve is increasing more rapidly when b is positive and going away from zero. It is

decreasing more rapidly when b is negative and going away from zero.