How to find the rule of an exponential function from two points and an asymptote

The rule you are looking for

The standard form of an exponential function has four parameters and a base:

When a question gives you the asymptote and two points, it is asking for the shorter form

Nothing is lost. The parameters $b$ and $h$ can always be folded into $a$ and $c$: for example, $3(2{)}^{2(x-1)}=3(4{)}^{x-1}=\frac{3}{4}(4{)}^x$. That leaves three unknowns, and the question hands you exactly three facts.

• $k$ is the height of the horizontal asymptote $y=k$. The curve gets closer and closer to it and never reaches it.
• $c$ is the multiplier. Each time $x$ goes up by 1, the vertical distance between the curve and the asymptote is multiplied by $c$.
• $a$ is that distance at $x=0$, with a sign: positive when the curve is above the asymptote, negative when it is below. That is why $f(0)=a+k$.

The method

1. Read $k$ from the asymptote. An asymptote $y=k$ means the rule ends in $+k$.
2. Subtract $k$ from the $y$-coordinate of each point. What is left behaves like a pure power of $c$: $$y_1-k=a(c{)}^{x_1}{y}_2-k=a(c{)}^{x_2}$$
3. Divide the second equation by the first. The $a$ cancels, and the exponent law for a quotient of powers leaves a single power: $$\frac{y_2-k}{y_1-k}=c^{x_2-x_1}$$
4. Take the root. $c$ is the positive number whose $(x_2-x_1)$th power is that ratio. A base is never negative, so even an even root gives only one answer.
5. Substitute $c$ back into either equation to find $a$, then check the finished rule on both points.

Why divide rather than subtract? Subtracting the two equations leaves $a$ in both terms. Once $k$ is gone, each side is a product, and a factor shared by the top and the bottom of a quotient cancels.

Worked example 1: a curve above its asymptote

Find the rule of the exponential function with asymptote $y=-7$ that passes through $(1,5)$ and $(3,101)$.

1. The asymptote gives $k=-7$, so $f(x)=a(c{)}^x-7$.
2. Subtracting $-7$ means adding 7: $5+7=12$ and $101+7=108$. So $$a(c{)}^1=12a(c{)}^3=108$$
3. Divide the second by the first: $$\frac{a(c{)}^3}{a(c{)}^1}=\frac{108}{12}\Rightarrow c^2=9\Rightarrow c=3$$
4. Back-substitute: $3a=12$, so $a=4$.
5. The rule is $f(x)=4(3{)}^x-7$. Check: $f(1)=12-7=5$ and $f(3)=4(27)-7=101$.

Worked example 2: reading the rule off a graph

The graph below shows an exponential function. Find its rule.

1. From the graph, $k=10$, so $f(x)=a(c{)}^x+10$.
2. Subtract 10: $-6-10=-16$ and $8-10=-2$. So $$a(c{)}^{-1}=-16a(c{)}^2=-2$$
3. Divide. The exponent is the horizontal distance between the points, $2-(-1)=3$: $$\frac{a(c{)}^2}{a(c{)}^{-1}}=\frac{-2}{-16}\Rightarrow c^3=\frac{1}{8}\Rightarrow c=\frac{1}{2}$$
4. Back-substitute: $a{\left(\frac{1}{2}\right)}^{-1}=2a=-16$, so $a=-8$.
5. The rule is $f(x)=-8{\left(\frac{1}{2}\right)}^x+10$. Check: $f(2)=-8\left(\frac{1}{4}\right)+10=8$.

Both differences were negative, so $a$ came out negative and the curve sits below its asymptote, with range $]-\infty ,10[$. The base is less than 1, yet the function increases: the gap between the curve and the asymptote is halved at every step, and the curve closes that gap from below.

Three checks before you trust the answer

• The two differences $y-k$ have the same sign. Since $c^x>0$ for every $x$, the term $a(c{)}^x$ always has the sign of $a$, so both points sit on the same side of the asymptote. If your differences have opposite signs, no rule of this form passes through both points: reread the asymptote.
• The ratio says which way the gap goes. With the points ordered so that $x_2>x_1$, a ratio greater than 1 gives $c>1$, and a ratio between 0 and 1 gives $0<c<1$.
• The rule reproduces both points. Substituting them takes two lines and catches a wrong root or a sign slip in $k$.

When the root is not a whole number

Suppose the division gives $c^4=5$. Then $c=\sqrt[4]{5}$, which is an exact answer and a perfectly good base. If you would rather have a whole-number base, the parameter $b$ provides one, because ${\left(\sqrt[4]{5}\right)}^x=(5{)}^{x/4}$. The rule can then be written $f(x)=a(5{)}^{x/4}+k$, which is the standard form with $b=\frac{1}{4}$ and $h=0$.

Check your understanding

No calculator. Every rule has the form $f(x)=a(c{)}^x+k$, with $a\ne 0$, $c>0$ and $c\ne 1$, unless the question says otherwise.

1. The function $f(x)=a(c{)}^x$ has the $x$-axis as its asymptote and passes through $(-1,2)$ and $(2,54)$. Find $a$ and $c$.
2. The asymptote is $y=1$, and the curve passes through $(0,33)$ and $(5,2)$. Find the rule.
3. Find the error. To find the rule with asymptote $y=4$ through $(1,10)$ and $(2,22)$, a student divides 22 by 10 and takes $2.2$ as the base. What went wrong? Find the correct rule.
4. True or false: some function of this form has the asymptote $y=5$ and passes through $(0,3)$ and $(1,8)$. Justify.
5. True or false: some function of this form passes through both $(1,7)$ and $(4,7)$. Justify.
6. The asymptote is $y=-1$, and the curve passes through $(1,-7)$ and $(3,-55)$. Find the rule, say whether $f$ is increasing or decreasing, and give its range.
7. The asymptote is $y=1$, and the curve passes through $(3,7)$ and $(6,19)$. Find the rule, then write it in the form $f(x)=a(3{)}^{bx}+k$.
8. News of a snow day spreads through a school of 800 students. The number of students who have heard it $t$ hours after it breaks is modelled by $N(t)=a(c{)}^t+k$, with asymptote $N=800$. After 1 hour, 400 students have heard; after 3 hours, 700 have. Find the rule, then find when 775 students have heard.

For more practice, the free Secondary 4 Exponential Functions worksheet has questions on finding a rule from two points, graphing, and half-life models, with each method explained on the page. When the unknown is the exponent rather than the base, the tool you need is the logarithm, and the Logarithmic Functions worksheet picks up from there.

Practise with a free worksheet

• Secondary 4 Exponential Functions worksheet
• Secondary 4 Logarithmic Functions worksheet
• Secondary 4 Arithmetic worksheet (laws of exponents)