Question

What effect will introducing a constant have on an exponential function?

Answer 1

.To really see what happens visually, go to fooplot.com and enter the base function y = e^x to look at its graph (In the textedit field below the label "Function y(x)", just enter "e^x" and hit the "Enter" key on your keyboard ... you don't need to enter the "y="). Then type in each of the example functions provided below to see what happens based upon the constant introduced.



y = e^x is an exponential function that increases gradually from (-âž, 0) until it reaches the point (0,1). Thereafter, it very sharply increases on (0,âž).



.....If you introduce a constant A > 1,

.....y = Ae^x will increase gradually from (-âž, 0) until it reaches the point (0,A).

.....Thereafter, it will rise faster than the base function y = e^x

.....Example. y = 5e^x



.....If your constant A is less than 1 but still positive: 0 < A < 1,

.....y = Ae^x will increase gradually from (-âž, 0) until it reaches the point (0,A).

.....Thereafter, it will have a slower rise on (0,âž) than the base function y = e^x.

.....Example. y = (1/5) e^x



.....If your constant A = -1,

.....y = Ae^x will look like y = Ae^x when A=1 except that the graph will be flipped over the x-axis.

.....Instead of increasing gradually on (-âž, 0), it will decrease gradually on (-âž, 0)

.....until it passes through the point (0, -1) and then veer downward on (0,âž)

.....Example. y = -e^x



.....If your constant A is between -1 and 0: -1 < A < 0,

.....y = Ae^x will be flipped over the x-axis and decrease gradually from(-âž, 0)

.....until it reaches the point (0,A).

.....Thereafter, it will have a slower decrease on (0,âž)

.....Example. y = (-1/5) e^x



.....If your constant A is less than 0: A < -1,

.....y = Ae^x will be reflected (flipped) over the x-axis and decrease gradually from (-âž, 0)

.....until it reaches the point (0,A). Thereafter, it will sharply veer downward.

.....Example. y = -5 e^x



y = e^x + A will translate the base function y = e^x by A units. If A > 0, it will essentially shift the graph upward by A. If A < 0, it will shift the graph downward by A.

.....Example. y = e^x + 5

.....Example. y = e^x - 5