Question

Consider the following properties of the graph of an exponential function.
a. Horizontal Asymptote at x=-2
b. The graph is increasing from (-infinity, infinity)
c. The x-intercept is at (1,0)
What is the equation for this graph?

Answer 1

The parent exponential function is:

$y= e^{x}$

The Horizontal asymptote of parent exponential function is y =0. For the function to have asymptote y=-2, it must be shifted 2 units down. So resulting graph will be:

$y=e^{x} -2$

The function is increasing from left to right.

The above function does not have x-intercept at (1,0). For the function to have an x-intercept at (1,) it must be multiplied with some co-efficient as shown below:

$y=a e^{x}-2$

For x=1, y=0. So we can write:

$0=a e^{1}-2 \\ \\ 2=ae \\ \\ a= \frac{2}{e}$

So, the exponential function satisfying the given conditions will be:

$y= \frac{2}{e} e^{x}-2 \\ \\ y=2 e^{x-1} -2$