4.8 Fitting exponential models to data  (Page 9/12)

Use a graphing utility to find an exponential regression formula $f(x)$ and a logarithmic regression formula $g(x)$ for the points $\left(1.5,1.5\right)$ and $\left(8.5,\text{ 8}\text{.5}\right).$ Round all numbers to 6 decimal places. Graph the points and both formulas along with the line $y=x$ on the same axis. Make a conjecture about the relationship of the regression formulas.

Verify the conjecture made in the previous exercise. Round all numbers to six decimal places when necessary.

Find the inverse function $f^{-1}\left(x\right)$ for the logistic function $f(x)=\frac{c}{1+a{e}^{-bx}}.$ Show all steps.

Use the result from the previous exercise to graph the logistic model $P(t)=\frac{20}{1+4e^{-0.5t}}$ along with its inverse on the same axis. What are the intercepts and asymptotes of each function?

Determine whether the function $y=156{\left(0.825\right)}^t$ represents exponential growth, exponential decay, or neither. Explain

The population of a herd of deer is represented by the function $A(t)=205{(1.13)}^t,$ where $t$ is given in years. To the nearest whole number, what will the herd population be after $6$ years?

Find an exponential equation that passes through the points $\text{(2, 2}\text{.25)}$ and $(5,60.75).$

Determine whether [link] could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.

A retirement account is opened with an initial deposit of $8,500 and earns $8.12\%$ interest compounded monthly. What will the account be worth in $20$ years?

Hsu-Mei wants to save $5,000 for a down payment on a car. To the nearest dollar, how much will she need to invest in an account now with $7.5\%$ APR, compounded daily, in order to reach her goal in $3$ years?

Does the equation $y=2.294e^{-0.654t}$ represent continuous growth, continuous decay, or neither? Explain.

Suppose an investment account is opened with an initial deposit of $\text{\$10,500}$ earning $6.25\%$ interest, compounded continuously. How much will the account be worth after $25$ years?

Graph the function $f(x)=3.5{\left(2\right)}^x.$ State the domain and range and give the y -intercept.

Graph the function $f(x)=4{\left(\frac{1}{8}\right)}^x$ and its reflection about the y -axis on the same axes, and give the y -intercept.

The graph of $f(x)={6.5}^x$ is reflected about the y -axis and stretched vertically by a factor of $7.$ What is the equation of the new function, $g(x)?$ State its y -intercept, domain, and range.