Exponential growth and decay

Compare the graphs of $y = 2^x$ and $y = 2^{-x}$. Since $2^{-x} = \left( \dfrac{1}{2} \right)^x$, the first grows as $x$ increases while the second decays.

Symmetry about the $y$-axis

The two graphs are symmetric about the $y$-axis: replacing $x$ with $-x$ swaps them. Both pass through $y = 1$ at $x = 0$, meeting at $(0, 1)$.

Reciprocals of each other

Multiplying the two values at any $x$ gives $2^x \cdot 2^{-x} = 2^0 = 1$, always. The functions are reciprocals of each other: when one doubles, the other halves.

Equal steps, equal factors

The essence of an exponential is that equal steps multiply by equal factors. On the growing graph each unit step in $x$ doubles the value; on the decaying graph each unit step halves it. The doubling time, or the half-life, is the same length wherever it is measured. The factor does not care how large or small the quantity has already become.

Inverting the base and negating the exponent

The curve $y = 2^{-x}$ is $y = 2^x$ reflected in the $y$-axis, and it is equally $y = \left( \dfrac{1}{2} \right)^x$, the exponential with the reciprocal base. Inverting the base and negating the exponent are the same operation, which is what the law $a^{-x} = \left( \dfrac{1}{a} \right)^x$ asserts; on the graph, that law appears as a reflection in the $y$-axis.

Properties they share

Both functions have range $y > 0$ and take the $x$-axis as an asymptote. The growing curve approaches $0$ toward the left and the decaying curve approaches it toward the right, but neither ever arrives.

Applications

The large dots mark the crossing $(0, 1)$ and the mirror-image points $(1, 2)$ and $(-1, 2)$.