Exponential Functions

The Exponential Function

Compare $x^2$ with $2^x$. In $x^2$ the variable is the base and the exponent is fixed. In $2^x$ the base is fixed and the variable is the exponent. Only $2^x$ is exponential.

Every exponential $f(x)=a^x$ passes through $(0,1)$, because $a^0=1$. It also passes through $(1,a)$, because $a^1=a$. Drag the slider to change the base and watch those two points.

One base is special: the irrational number $e\approx 2.71828\ldots$. The function $f(x)=e^x$ is the "natural" exponential, and it shows up in continuous growth, radioactive decay and cooling. Write $e$ exactly, and use a decimal only when a problem asks for one.

To evaluate an exponential you need two rules for exponents: $a^{-n}=\dfrac1{a^n}$ and $a^{m/n}=\left(\sqrt[n]{a}\right)^m$.

When you know two points of $f(x)=k\cdot a^x$, you can find both $k$ and $a$.

Growth and Decay

The reason is the growth factor: going from $x$ to $x+1$ multiplies the output by $a$ every time, since $\dfrac{a^{x+1}}{a^x}=a$. A factor above $1$ makes each step bigger. A factor below $1$ makes each step smaller.

Notice that $\left(\tfrac12\right)^x=2^{-x}$. The graph of $2^{-x}$ is the graph of $2^x$ reflected across the $y$-axis. This leads to two reflection rules.

Use these clues to match a graph with its equation:

• Where does it cross the $y$-axis? Through $(0,1)$ means $a^x$. Through $(0,-1)$ means $-a^x$.

• Is it rising or falling as you move right? Rising means base above $1$. Falling means base below $1$, or a negative sign in front.

• Among growing curves, the one with the larger base is steeper to the right of the $y$-axis and flatter to the left.

Try it: the curves below are $2^x$, $3^x$, $0.5^x$ and $-2^x$, the same four functions as in the matching exercise.

Transformations, Asymptotes and Intercepts

Now shift and stretch the graph. Take $g(x)=A\cdot b^{\,x-h}+k$:

• $x-h$ in the exponent moves the graph right $h$ units. A plus sign, as in $x+2$, moves it left.

• $+k$ at the end moves the graph up $k$ units.

• The asymptote moves with the graph: it becomes $y=k$.

• A negative $A$ reflects the graph over the horizontal line $y=k$.

Why does the asymptote become $y=k$? If $b>1$, then as $x\to-\infty$ the term $b^{x-h}\to 0$, so $g(x)\to k$. If $0

Slide $p$ and $q$ and watch the asymptote and the graph move.

A handy trick: the key points $(0,1)$ and $(1,2)$ of $2^x$ move to $(h,\,1+k)$ and $(h+1,\,2+k)$.

For intercepts: the $y$-intercept is $g(0)$. For an $x$-intercept, set $g(x)=0$ and solve. In the next section you'll see how to solve these equations in general. For now, rewrite both sides with the same base.

Equations: Same Base and Common Factor

An exponential equation has the unknown in an exponent. Two techniques solve many of them. Both rely on the power rules $a^{m+n}=a^m a^n$, $a^{m-n}=\dfrac{a^m}{a^n}$ and $(a^m)^n=a^{mn}$.

See it on a graph: the line $y=9$ crosses $y=3^x$ exactly once, at $x=2$.

Equations: Logarithms and Substitution

Sometimes you cannot reach a common base: $2^x=5$ has no easy answer. That is what logarithms are for.

Two facts are enough for this page:

To solve an exponential equation: isolate the power, take $\ln$ of both sides, bring down the exponent, then solve.

Substitution. If an equation has both $a^{2x}$ and $a^x$, it is a quadratic in disguise. Since $a^{2x}=(a^x)^2$, let $t=a^x$ (with $t>0$), solve the quadratic in $t$, then go back to $x$.

Systems. If the unknowns sit in exponents of different bases, substitute $u=a^x$ and $v=b^y$. You get an ordinary linear system, which you can solve by elimination.

The graph above is $y=4^x-2^x-6$. It crosses the $x$-axis only once, at $x\approx1.585$. The impossible $t=-2$ never produces a second crossing.

Models of Growth and Decay

Any quantity that changes by the same factor in equal time periods follows $$f(t)=a\cdot b^{\,t},$$ where $a$ is the starting amount (the $y$-intercept) and $b$ is the growth factor. To get $b$ from a percent change $r$ (written as a decimal): $b=1+r$ for growth and $b=1-r$ for decay. A $15\%$ increase has $b=1.15$, and a $15\%$ decrease has $b=0.85$.

Try it: change the numbers and see how interest compounds.

Back to the start: Plan A (adding $300$ views per hour) gives $200+300t$, which is $3{,}200$ views after $10$ hours. Plan B (doubling) gives $V(t)=200\cdot2^t$, which is $200\cdot1024=204{,}800$ views after $10$ hours. Repeated multiplication beats repeated addition by a huge margin.

More Models: Depreciation, Cooling and Logistic Growth

Some models need logarithms to find when something happens.

Newton's law of cooling. An object's temperature moves toward the room temperature $T_a$ according to $$T(t)=T_a+(T_0-T_a)\,e^{-kt},$$ where $T_0$ is the starting temperature and $k>0$ is the cooling constant. The exponential part $e^{-kt}\to0$ as $t\to\infty$, so $T(t)\to T_a$. The line $y=T_a$ is a horizontal asymptote.

Logistic growth. Real populations cannot grow forever. A logistic model $$P(t)=\frac{L}{1+Ce^{-kt}}$$ starts like an exponential, then levels off at the limit $L$. As $t\to\infty$, $e^{-kt}\to0$ and $P\to L$, so $y=L$ is a horizontal asymptote.

Domain, Sign and Intersections

Exponentials combine with other expressions, so the usual function-analysis tools apply. One fact does most of the work: $a^{g(x)}>0$ for every $x$. An exponential factor is never zero and never negative.

Asymptotes and Continuity

How an exponential behaves at the ends tells you its horizontal asymptotes.

So for $f(x)=e^{g(x)}$:

• Horizontal asymptote: if $g(x)\to L$ as $x\to\pm\infty$, then $f(x)\to e^L$, and $y=e^L$ is a horizontal asymptote.

• Vertical asymptote: if $g(x)\to+\infty$ as $x\to c$ from one side, then $f(x)\to\infty$ there, so $x=c$ is a vertical asymptote. If $g(x)\to-\infty$ instead, then $f(x)\to0$ and there is no asymptote on that side.

Continuity. A piecewise function is continuous at a join $x=c$ when $$\lim_{x\to c^-}f(x)=\lim_{x\to c^+}f(x)=f(c).$$ Exponentials and polynomials are continuous on their own, so you only need to check the places where the pieces meet.

Summary