Exponential Functions Formulas and Notes

Exponential Functions

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These are functions in the form \displaystyle y=a \cdot b^{x}+k.
When \displaystyle a > 0, the function increases.
When \displaystyle a < 0, the function decreases.

When \displaystyle b > 1, the function increases.
When \displaystyle 0 < b < 1, the function decreases.
When \displaystyle b \leq 0, the function is undefined.

\displaystyle k is the horizontal asymptote. When \displaystyle k > 0, the function shifts vertically upwards by k units.
When \displaystyle k < 0, the function shifts vertically downwards by k units.
Example
Plot \displaystyle f(x) = 4^{x}.
\displaystyle f(x) = 1 \cdot 4^{x}.
\displaystyle a = 1, b = 4
\displaystyle a > 0, b \geq 1. The function increases.

x	-2	-1	0	1	2
f(x)	0.0625	0.25	1	4	16

Example
Plot \displaystyle f(x) = -1 \cdot 4^{x}.
\displaystyle a = -1, b = 4
\displaystyle a < 0. The function decreases.

x	-2	-1	0	1	2
f(x)	-0.0625	-0.25	-1	-4	-16

Example
Plot \displaystyle f(x) = 4^{x} + 3.
\displaystyle f(x) = 1 \cdot 4^{x} + 3.
\displaystyle a = 1, b = 4, k = 3
\displaystyle a > 0; k > 0. The function increases. The function shifts vertically upwards by 3 units.

x	-2	-1	0	1	2
f(x)	3.0625	3.25	4	7	19

Example
Plot \displaystyle f(x) = 4^{x} - 3.
\displaystyle f(x) = 1 \cdot 4^{x} - 3.
\displaystyle a = 1, b = 4, k = - 3
\displaystyle a > 0; k < 0. The function increases. The function shifts vertically downwards by 3 units.

x	-2	-1	0	1	2
f(x)	-2.9375	-2.75	-2	1	13