Graphs of Exponential Functions

Compare $y=2x+1$ with $y=2^x$ in a table with $x$ going up by 1. The linear function goes $1,3,5,7,9$: every step adds 2, so the first differences are constant. The exponential goes $1,2,4,8,16$: its first differences are $1,2,4,8$ (not constant, and growing), but every step multiplies by 2, so the ratios $\frac{y_{n+1}}{y_n}$ are constant. The reason is the product law: $b^{x+1}=b^x\cdot b$, so moving one unit right always multiplies the output by $b$. Notice that the differences of an exponential are themselves exponential, which is why "check the differences" never settles down for these functions.

Linear vs. exponential growth: from data | High School Math | Khan Academy

For $y=b^x$ with $b>0$ and $b\ne 1$: if $b>1$, the function is increasing ("growth") — larger $x$ always gives a bigger output. If $0<b<1$, the function is decreasing ("decay") — larger $x$ gives a smaller (but still positive) output. $b=1$ is excluded because $1^x=1$ for every $x$, which is just a constant, not genuinely exponential. Negative bases are excluded too: ${(-2)}^x$ flips sign at every whole number and ${(-2)}^{1/2}=\sqrt{-2}$ is not even real, so its "graph" is a scatter of disconnected points. Watch for decay written in disguise: $y=2^{-x}$ equals ${\left(\frac{1}{2}\right)}^x$, so it is decay even though the base you see is 2.

• Always passes through $(0,1)$, since $b^0=1$ for any valid base. Two more easy points are $(1,b)$ and $\left(-1,\frac{1}{b}\right)$, enough for a quick sketch.
• Horizontal asymptote at $y=0$ — the graph gets arbitrarily close to the $x$-axis but never touches or crosses it, because $b^x>0$ for every real $x$. So there is no $x$-intercept.
• Domain: all real numbers, $x\in \mathbb{R}$.
• Range: $y>0$, i.e. $(0,\infty )$ — never zero, never negative.
• Always increasing ($b>1$) or always decreasing ($0<b<1$), so it never turns around and passes the horizontal line test.

For small $x$, a polynomial like $x^2$ can look bigger than an exponential like $2^x$. But exponential growth eventually outpaces every polynomial, no matter how large its degree — because an exponential multiplies by the same factor at every step, while a polynomial's growth rate slows down relatively as $x$ increases. Going from $x=10$ to $x=11$, $x^2$ grows by about 21% while $2^x$ grows by 100%, and that 100% never shrinks. In the table, $x^2$ and $2^x$ tie at $x=2$ and $x=4$, and after that the exponential stays ahead for good. This is why early growth is so easy to underestimate in real life: a case count that doubles every three days looks harmless for the first couple of weeks and then can overwhelm a hospital system within a month, and a savings account earning compound interest looks almost linear for years before the curve bends sharply upward.

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