Graphs of Exponential Functions

Think of the basic exponential curve $y = b^x$ as a rocket launch: it starts low on the left, passes through the point $(0, 1)$, and shoots upward to the right. There's an invisible floor at $y = 0$ that the curve gets close to but never touches — that floor is called the horizontal asymptote.

Now imagine you can grab that curve and slide it around. Shifting it up or down (adding or subtracting a number at the end, like $+3$ or $-5$) moves the invisible floor too. Shifting it left or right (changing what's in the exponent, like $x - 2$) slides the whole picture sideways but keeps the floor at the same height. You can also flip the curve upside down by putting a negative sign in front.

The domain (which $x$-values you can plug in) is always all real numbers — exponentials accept any input. The range (which $y$-values come out) depends on where that floor is and whether you've flipped the curve. If the floor is at $y = 3$ and the curve sits above it, the range is everything above 3. If the curve is flipped below the floor, the range is everything below it.

The one rule that trips people up: only the number added or subtracted *outside* the exponent moves the asymptote. A shift inside the exponent (like $x - 3$) just slides the curve left or right — the floor stays put. So when you see $2^{x-3} + 5$, the asymptote is $y = 5$, not anything to do with the 3.