03 Exponential Functions Free Online FlashCards

What is the standard form of an exponential function?

An exponential function has the form $f(x)=ab^x$, where $a$ is the initial value or vertical scale factor and $b$ is the growth or decay factor.

How does the base determine exponential behavior?

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

What are key graph features of $f(x)=ab^x$?

For $f(x)=ab^x$ with $a>0$, the domain is all real numbers, the range is positive real numbers, the horizontal asymptote is $y=0$, and the $y$-intercept is $(0,a)$.

How do linear and exponential models differ?

A linear function changes by a constant amount over equal intervals, whereas an exponential function changes by a constant percentage or factor.

What model represents discrete exponential growth?

For growth by rate $r$ per period, use $A(t)=A_0(1+r)^t$, where $A_0$ is the initial amount.

How is a percentage rate converted into a factor?

A 6% increase uses $r=0.06$ and growth factor $1.06$. A 6% decrease uses decay factor $1-0.06=0.94$.

What is the continuous exponential model?

Continuous change is modeled by $A(t)=A_0e^{kt}$. A positive $k$ indicates continuous growth, while a negative $k$ indicates continuous decay.

What formula gives continuous doubling time?

For continuous growth, the doubling time is $t_d=\frac{\ln 2}{k}$, where $k>0$.

What formula gives continuous half-life?

For continuous decay, the half-life is $t_{1/2}=\frac{\ln 2}{|k|}$, where $k<0$.

What is the compound-interest formula?

For interest compounded $n$ times per year, use $A(t)=P\left(1+\frac{r}{n}\right)^{nt}$, where $P$ is principal and $r$ is the annual decimal rate.

How can an exponential equation be solved using a common base?

Rewrite both sides with the same positive base, then set the exponents equal. This uses $b^u=b^v\Rightarrow u=v$ for $b>0$ and $b\ne 1$.

How do logarithms solve $b^x=c$?

If $b^x=c$, then $x=\log_b(c)=\frac{\ln c}{\ln b}$. Taking logarithms is useful when the bases cannot conveniently be matched.