# "e"-Motion

Recall the graph of the exponential function y=e^x.

This is a graph of the equation y=ae^(bx)+c, with a, b, and c all equal to 1.

Let's observe how the graph changes as the values of a, b, and c change.

### Changing a:

Looking at the examples above we see that the value of a stretches or shrinks the function.

When the value of a is greater than 1, the y values of the function are increased, which "stretches" the graph vertically.

When a is less than 1, the y values of the function are decreased, which "shrinks" the graph vertically.

The value of a also changes the y-intercept of the function.

### Changing b:

Looking at the examples above we see that as the value of b affects the rate of change of the function.

If the value of b is greater than 1 the rate of change increase. If the value of b is less than one the rate of change decreases.

We see that this change in rate does not effect the y-intercept at all.

### Changing c:

Looking at the examples above we see that as the value of c changes the graph is shifted vertically.

The asymptote of our original function, y=e^x, is x=0.

We see as the value of c changes the asymptote is shifted. The asymptote of y=e^x+2 is x=2, and the asymptote of y=e^x+(1/2) is x=(1/2).

Though the asymptotes and y-intercepts of the graphs have shifted, the rate is the same for all three functions.

### Conclusion

Using the examples above we can better determine what the graph of an exponential function will look like prior to graphing it.

Using y=e^x as a reference, we will be able to determine the y-intercept, the rate of change, a vertical stretch or shrink, or vertical shifts using the values of a, b, and c.

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