Assignment 1
by
Amanda Newton
Exponential Importance
Exponential graphs with base e can be modeled by various applications. The equation has 3 constants that we should consider, but first, let's examine the parent graph 
where a=1, b=1, and c=0 or 
:
We can see that as the value of x increases, y increases at a rapid rate. As x decreases, the value of y decreases as well but at a slower rate approaching zero. This is a characteristic of exponential equations. We should keep in mind that though e is a very special and interesting number, it is in fact a number, not a variable. The value of e is irrational, in common language, it cannot be written as a fraction. The value is a decimal that does not repeat itself or terminate. There are well over 10,000 digits of e recorded, but we will estimate the number at 2.718. If 2<e<3 then let's look at the graph of each and compare. We would assume the graph with base 3 will look most like the parent graph since the value is closer to 3.
We can see that the graph with base e looks more like the graph with base 3 than that with base 2. Now that we have seen the parent graph and how it looks in comparison to other exponential functions, let's examine properties of the exponential graph with base e. First we will look at how the graph changes as the value of the constant "c" in the equation
. We could predict from our Algebra knowledge that the graph will be shifted up one for every one increase of c. "Shifted up" refers more directly to the y-intercept, so the the shape of the graph is preserved by the y-intercept will increase as c increases. Since the parent graph has a y-intercept of 1, we might make the hypothesis that the y-intercept will be c+1. The graphs below confirms our predictions. The asymptote however, increases by 1 for every 1 increase in c. The parent graph or when c=0 has an asymptote of y=0, but when c=1, the asymptote is y=1. This is because the only way the value of y=0 is when a=0 and c=0. The graph will otherwise never intersect the x-axis.
Now looking back at our equation we can see that the y-intercept when the coefficient of e is 1 will always be c+1. The y-intercept comes from the equation when x=0. We are dealing with an exponential equation where anything to the power of 0 is 1. So why then does the coefficient of e have to be 1? Let's look at the graph when a=2 and c=1.
The y-intercept increased by 1 in relation to our previous equation where a=1. Why and how could this be? As said before, anything to the power of 0 is 1. So we when looking for the y-intercept in this exponential equation, we are really looking at the equation y=a(1)+c, where y represents our y-intercept, a represents the coefficient of e, and c represents the constant of the equation. In the graph above, the y-intercept appears to be 3. Let's use our new equation to check: y=2(1)+1=3. 3 is indeed the y-intercept and we see that the equation y=a+c generates this y-intercept. But does the value of "a" change the shape of the graph or does it only affect the y-intercept of the graph?
Looking at the left side of the graph, it is apparant that increasing the value of "a" does not change the asymptote y=0. However, as "a" increases, the y-value increases at a greater rate. We would expect the opposite to be true when the value of "a" decreases. When a=0, we know the graph will be a horizontal line through "c": y=0(e^x)+c therefore y=c. But what about when a<0?
We can see that the opposite does indeed hold when c<0. Taking the opposite of "c" reflects the graph over the x-axis, but the asymptote of y=0 is not affected. In the cases above c=0, but the case when c does not equal zero would be another great exploration for students and teachers alike could try, but for general cases we will look at the constants seperately. Another great extension would be to look at non-integer values, but when we want to generalize the effects of the constants, it may be easier and more clear to start off with the "basics."
Let's not forget about our b value! Our "b" is the coefficient of the exponent x. What if b>1?
We can see that the y-intercept is not changed, nor has the asymptote of y=0, but the y value increases much more rapidly in the cases where b>1. The change in the graph where b=1 and b=2 is much more drastic than when b=3 versus when b=4. The graph is still changed, but certainly not as much. We can expect this relationship to hold, so the graph when b=5 will look even more like the graph when b=4.
Different combinations of the 3 constants a, b, and c, will yield different results. These can be investigated and explored in depth more thoroughly. A great way to get students involved and conceptualize exponential problems is to have them investigate these graphs and pose application problems that represent e base exponential equations. As with any mathematics topic, there is always more to explore.
