Explores transformations (shifts, stretches, reflections) of exponential graphs and common real-world applications like compound interest and continuous growth models.
What is the formula for continuously compounded interest?
$$ A = Pe^{rt} $$
How do \(h\) and \(k\) transform the graph of \(y=b^x\) in the function \(y=a b^{x-h} + k\)?
\(h\) causes a horizontal shift (right for \(h>0\), left for \(h<0\)). \(k\) causes a vertical shift (up for \(k>0\), down for \(k<0\)) and moves the horizontal asymptote to \(y=k\).
How does a negative sign in front of \(a\) in \(y = ab^x\) affect the graph?
A negative \(a\) reflects the graph across the x-axis. For example, the graph of \(y = -2^x\) is a reflection of \(y = 2^x\) across the x-axis.
How does a negative sign on the exponent, as in \(y = b^{-x}\), affect the graph?
A negative exponent reflects the graph across the y-axis. This is because \(y = b^{-x} = (b^{-1})^x = (1/b)^x\), which changes a growth function into a decay function and vice versa.
What is the formula for compound interest, where interest is compounded \(n\) times per year?
$$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$
In the compound interest formula, what do \(P\), \(r\), \(n\), and \(t\) represent?
\(P\) is the principal (initial amount), \(r\) is the annual interest rate (as a decimal), \(n\) is the number of times interest is compounded per year, and \(t\) is the time in years.
In the continuously compounded interest formula, what do \(P\), \(r\), and \(t\) represent?
\(P\) is the principal (initial amount), \(r\) is the annual interest rate (as a decimal), and \(t\) is the time in years. \(e\) is the natural base.
If you invest \(\$1000\) at an annual rate of \(5\%\) compounded quarterly, what is the expression for the amount after \(10\) years?
$$ A = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \cdot 10} $$
A substance has a half-life of \(20\) days. If you start with \(100\) grams, what is the exponential model for the amount \(A\) remaining after \(t\) days?
The model is \(A(t) = 100 \left(\frac{1}{2}\right)^{t/20}\). The initial amount is \(100\)g, the decay factor is \(1/2\), and the exponent is scaled by the half-life period.
Write the general function \(A(t)\) for exponential growth or decay.
The function is \(A(t) = A_0 (1+r)^t\) for growth or \(A(t) = A_0 (1-r)^t\) for decay, where \(A_0\) is the initial amount and \(r\) is the rate of change per time period \(t\).
