[4.2] Exponential vs. Linear Functions

class 17


Basic concepts

A function is linear if it changes at a constant absolute rate. $$y(x)=b+mx$$

A function is exponential if it changes at a constant percentage rate. $$y(x)=ab^x \text{ or }=a(1+r)^x$$ where $b=(1+r)$.

Properties of exponentials

  • $$b^0=1$$
  • $$b^1=b$$
  • $$b^3=b*b*b$$
  • $$b^{-1}=\frac 1b$$
  • $$b^{-3}=\frac{1}{b*b*b}$$
  • $$b^xb^y=b^{x+y}$$ E.g. $32=4*8=2^2*2^3=2^5=32.$
  • $$b^x/b^y=b^{x-y}$$ E.g. $8/4=2^3/2^2=2^{3-2}=2^1=2.$
  • $$(b^x)^y=b^{xy}$$ E.g. $(2^3)^2=(8)^2=64=2^{3*2}=2^{6}.$
  • $$\sqrt{b}\equiv b^{\frac{1}{2}}.$$ Consider $\sqrt 9=3$. We could also write the square root of 9 as: $$9^{1/2}=(3^2)^{1/2}=3^{2*1/2}=3^1=3$$

Find the exponential equation

Find the exponential equation for the red graph, in the form $y(x)=ab^x$: Find $a$ and $b$.

  1. Find 2 points on the graph.
  2. Put the $x$ and $y$ values of the two points into the model exponential to get two equations: $$70=a b^{0.5}$$ and $$20=a b^{3}$$
  3. Divide one equation by the other equation! This eliminates $a$, and gives us one equation for $b$: $$\begineq \frac{70}{20}&=&\frac{ab^{0.5}}{ab^3}=b^{0.5-3}\\ 3.5&=&b^{-2.5} \endeq $$
  4. Solve this equation for $b$: $$\begineq (3.5)^{-1/2.5}=(3.5)^{-0.4}&=&(b^{-2.5})^{-1/2.5}\\ 0.606&=&b \endeq$$
  5. Put $b=0.606$ back into one of the model equations, and solve for $a$: $$20=a (0.606)^{3}=a 0.223$$ $$a=\frac{20}{0.223}=89.7$$

Now, you have $a$ and $b$. The equation of the red line is: $$\color{red}{y=89.7 (0.606)^x} =89.7(1-0.394)^x$$ We could also say that the function is decaying exponentially at a rate of -39.4%. ($r=-0.394$)

Now find an equation for the $\color{green}{\text{green}}$ function.

Table example


  • Find a formula for the function which is alway changing by the same absolute amount.
  • Find a formula for the function which is always changing by the same percentage.
  • What are the domains and ranges of the two functions you found?

In words

[Problem 4.2 #38] - There were 178.8 million licensed drivers in the U.S. in 1989, and 187.2 million licensed drivers in 1999.

Find a formula for the number of drivers, $N$, in the US, a number of years $t$ after 1989 assuming growth is exponential. What is the annual percentage rate of increase?