10/14 10. Exponential Functions and Applications Free Online FlashCards

What is the general form of an exponential function?

An exponential function has the form $f(x)=ab^x$, where $a\neq 0$, $b>0$, and $b\neq 1$.

How does the base determine exponential growth or decay?

If $b>1$, the function represents exponential growth. If $0<b<1$, it represents exponential decay.

How are percentage rates converted to exponential factors?

For a percentage growth rate $r$, use factor $1+r$. For a decay rate $r$, use factor $1-r$, with $r$ written as a decimal.

What are the y-intercept and horizontal asymptote of $f(x)=ab^x$?

For $f(x)=ab^x$, the y-intercept is $(0,a)$, because $f(0)=ab^0=a$. Without a vertical shift, the horizontal asymptote is $y=0$.

What model describes continuous growth or decay?

A continuous growth or decay model is $A(t)=A_0e^{kt}$, where $A_0$ is initial amount and $k$ is the continuous rate constant.

How much remains from 800 grams after three 6-year half-lives?

With a half-life of 6 years, 800 grams becomes $800(\frac{1}{2})^{18/6}=100$ grams after 18 years.

How can a growth factor be found from two values?

Substitute both values into $A(t)=A_0b^t$, then solve for $b$. For 2,000 becoming 2,420 in 3 years, $b=\sqrt[3]{1.21}\approx 1.065$.

What is the periodic compound-interest formula?

For principal $P$, annual rate $r$, periods per year $n$, and years $t$, use $A=P\left(1+\frac{r}{n}\right)^{nt}$.

What is the continuous compound-interest formula?

For continuous compounding, use $A=Pe^{rt}$, where $P$ is principal, $r$ is the annual rate, and $t$ is time in years.

How is an exponential equation solved using a common base?

Rewrite both sides with the same base, then set their exponents equal. For $3^{2x-1}=27=3^3$, solve $2x-1=3$ to get $x=2$.

What formula solves $a^x=c$ using logarithms?

For $a^x=c$, take logarithms and solve: $x=\frac{\ln c}{\ln a}$, provided $a>0$, $a\neq 1$, and $c>0$.

What steps solve an exponential equation with a linear exponent?

First isolate the exponential expression, take logarithms, bring down the exponent, solve the resulting linear equation, and check the result.