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Use a graphing utility to find an exponential regression formula $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ and a logarithmic regression formula $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ for the points $\text{\hspace{0.17em}}\left(1.5,1.5\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(8.5,\text{8}\text{.5}\right).\text{\hspace{0.17em}}$ Round all numbers to 6 decimal places. Graph the points and both formulas along with the line $\text{\hspace{0.17em}}y=x\text{\hspace{0.17em}}$ 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.
First rewrite the exponential with base e : $\text{\hspace{0.17em}}f(x)=1.034341{e}^{\text{0}\text{.247800x}}.\text{\hspace{0.17em}}$ Then test to verify that $\text{\hspace{0.17em}}f(g(x))=x,$ taking rounding error into consideration:
$\begin{array}{ll}g(f(x))\hfill & =4.035510\mathrm{ln}\left(1.034341{e}^{\text{0}\text{.247800x}}\text{\hspace{0.17em}}\right)-0.136259\hfill \\ \hfill & =4.03551\left(\mathrm{ln}\left(1.034341\right)+\mathrm{ln}\left({e}^{\text{0}\text{.2478}x}\text{\hspace{0.17em}}\right)\right)-0.136259\hfill \\ \hfill & =4.03551\left(\mathrm{ln}\left(1.034341\right)+\text{0}\text{.2478}x\right)-0.136259\hfill \\ \hfill & =0.136257+0.999999x-0.136259\hfill \\ \hfill & =-0.000002+0.999999x\hfill \\ \hfill & \approx 0+x\hfill \\ \hfill & =x\hfill \end{array}$
Find the inverse function $\text{\hspace{0.17em}}{f}^{-1}\left(x\right)\text{\hspace{0.17em}}$ for the logistic function $\text{\hspace{0.17em}}f(x)=\frac{c}{1+a{e}^{-bx}}.\text{\hspace{0.17em}}$ Show all steps.
Use the result from the previous exercise to graph the logistic model $\text{\hspace{0.17em}}P(t)=\frac{20}{1+4{e}^{-0.5t}}\text{\hspace{0.17em}}$ along with its inverse on the same axis. What are the intercepts and asymptotes of each function?
The graph of $\text{\hspace{0.17em}}P(t)\text{\hspace{0.17em}}$ has a y -intercept at (0, 4) and horizontal asymptotes at y = 0 and y = 20. The graph of $\text{\hspace{0.17em}}{P}^{-1}(t)\text{\hspace{0.17em}}$ has an x - intercept at (4, 0) and vertical asymptotes at x = 0 and x = 20.
Determine whether the function $\text{\hspace{0.17em}}y=156{\left(0.825\right)}^{t}\text{\hspace{0.17em}}$ represents exponential growth, exponential decay, or neither. Explain
exponential decay; The growth factor, $\text{\hspace{0.17em}}0.825,$ is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.$
The population of a herd of deer is represented by the function $\text{\hspace{0.17em}}A(t)=205{(1.13)}^{t},\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is given in years. To the nearest whole number, what will the herd population be after $\text{\hspace{0.17em}}6\text{\hspace{0.17em}}$ years?
Find an exponential equation that passes through the points $\text{\hspace{0.17em}}\text{(2,2}\text{.25)}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(5,60.75).$
$y=0.25{\left(3\right)}^{x}$
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.
x | 1 | 2 | 3 | 4 |
f(x) | 3 | 0.9 | 0.27 | 0.081 |
A retirement account is opened with an initial deposit of $8,500 and earns $\text{\hspace{0.17em}}8.12\%\text{\hspace{0.17em}}$ interest compounded monthly. What will the account be worth in $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years?
$\mathrm{\$}42,888.18$
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 $\text{\hspace{0.17em}}7.5\%\text{\hspace{0.17em}}$ APR, compounded daily, in order to reach her goal in $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}$ years?
Does the equation $\text{\hspace{0.17em}}y=2.294{e}^{-0.654t}\text{\hspace{0.17em}}$ represent continuous growth, continuous decay, or neither? Explain.
continuous decay; the growth rate is negative.
Suppose an investment account is opened with an initial deposit of $\text{\hspace{0.17em}}\text{\$10,500}\text{\hspace{0.17em}}$ earning $\text{\hspace{0.17em}}6.25\%\text{\hspace{0.17em}}$ interest, compounded continuously. How much will the account be worth after $\text{\hspace{0.17em}}25\text{\hspace{0.17em}}$ years?
Graph the function $\text{\hspace{0.17em}}f(x)=3.5{\left(2\right)}^{x}.\text{\hspace{0.17em}}$ State the domain and range and give the y -intercept.
domain: all real numbers; range: all real numbers strictly greater than zero; y -intercept: (0, 3.5);
Graph the function $\text{\hspace{0.17em}}f(x)=4{\left(\frac{1}{8}\right)}^{x}\text{\hspace{0.17em}}$ and its reflection about the y -axis on the same axes, and give the y -intercept.
The graph of $\text{\hspace{0.17em}}f(x)={6.5}^{x}\text{\hspace{0.17em}}$ is reflected about the y -axis and stretched vertically by a factor of $\text{\hspace{0.17em}}7.\text{\hspace{0.17em}}$ What is the equation of the new function, $\text{\hspace{0.17em}}g(x)?\text{\hspace{0.17em}}$ State its y -intercept, domain, and range.
$g(x)=7{\left(6.5\right)}^{-x};\text{\hspace{0.17em}}$ y -intercept: $\text{\hspace{0.17em}}(0,\text{7});\text{\hspace{0.17em}}$ Domain: all real numbers; Range: all real numbers greater than $\text{\hspace{0.17em}}0.$
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