Assignment 1 Write Up

Examine graphs of

First let's examine the graph of this exponential function as the constant "a" varies while keeping variables "b" and "c" constant. For this case, let's let b=1 and c=0:

So lets observe that as "a" varies in the equation , the y intercept varies proportionally. As the "a" increases, the y intercept of the exponential equational also increases. Matter of fact, the y-intercept actually equals the value of "a" in the exponential equation. To show this. let's observe when a=4:

How about when a=10:

From the previous graphs, we can see that the y-intercept equals the value of "a" in the exponential function . So as the variable "a" varies, the y-intercept varies in proportion.

Now let look at as the variable "b" varies while keeping "a" and "c" constant. For this case, let a=1 and c=0:

When we vary the constant "b" in our exponential equation, we can see that the y-intercept does change and stays at one. We have already discussed that the y-intercept equals the value of "a". In this case "a=1." Therefore, no matter what the value of "b" may be the intercept will be one. We have already seen that when "a" varies the y-intercepts varies, but when "b" varies the slope of the tangent line to the graph varies. Lets observe the following graph:

As "B" varies in the equation , the slope of the tangent (the red line above) varies as well. As "b" increases to infinity, the slope of the tangent also approaches infinity; getting closer and closer to the line x=0. When "b" decreases to zero, the slope of the tangent line approaches the line y=1 which is the y-intercept of both the exponential function and the linear function of the tangent line.

So we have explored the equation when the constants "a" and "b" vary. Now we need to observe the graph of the exponential function when the constant "c" varies. Quickly, let review what varies when the constant "a" and "b" change. When "a" varies in the equation, the intercept of the graph varies proportionally, actually the y-intercept= value of a. We have also concluded that when "b" varies, the slope of the tangent line at the point (1,1) varies. So lets observe the graph of the exponential function where the constant "c" is varying. For the purpose of this graph we are going to keep the constants a=1 and b=1.

Lets observe that when "c" varies, the slope of the tangent line (red line) at the point (1,1) stays constant at a slope of 1 and moves simultaneously with the exponential graph. The constant "c" varies the vertical shift of the graph. More or less, the horizontal asymptote (blue line) is varying as the constant "c" varies. Like most function graphs the constant "c" in the equation controls the vertical shift of the graph and in this equation the constant "c" plays the same role. Interestingly, the value of "c" in a exponential equation such as equals the horizontal asymptote of the graph represented by the blue line in the above animation.

Lastly, lets observe the the special cases of the equation : "What if a=0?"

The exponential part of the equation will disappear and we will get the horizontal line y=c. We can also explore the case: "What if b=0?": ( Then we get the graph of y=1+c because e^0 power is 1). So in conclusion, we have explored the graph as the constants "a", "b", and "c" vary and also what parts of the graph are varying as these constant vary. Hopefully this exploration gives the reader a better understanding of exponential equations and their graphs.