Question

write the exponential function. please show work

Answer 1

Answer:

The function can be considered to be a horizontal translation and a vertical reflection of a function that decays by a factor of 5/20 = 1/4 as x increases by 2.

.. y = -20(1/4)^((x +2)/2)

This can be simlified to

.. y = -5*(1/2)^x

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If you allow vertical translation, any number of growth or decay functions can be written that will have these two points on their curve. The graph shows a growth function that passes through the same two points.

Step-by-step explanation:

Answer 2

Answer:

$y=10e^{x\ln4}=10(4)^x$

$y=0.4e^{-x\ln3}=0.4(3)^{-x}$

Step-by-step explanation:

$\boxed{\begin{minipage}{9 cm}\underline{General form of an Exponential Function with base e }\\\\ y=ae^{kx} \\\\where:\\\phantom{ww} \bullet a is the initial value ( y -intercept). \\ \phantom{ww} \bullet e is Euler's number. \\ \phantom{ww} \bullet k is some constant.\\\end{minipage}}$

Given points:

• (1, 40)
• (3, 640)

Substitute the points into the formula to create two equations:

$\implies ae^{k}=40$

$\implies ae^{3k}=640$

Divide the equations to eliminate a and solve for k:

$\implies \dfrac{ae^{3k}}{ae^{k}}=\dfrac{640}{40}$

$\implies \dfrac{e^{3k}}{e^{k}}=16$

$\implies e^{2k}=16$

$\implies \ln e^{2k}=\ln 16$

$\implies \ln e^{2k}=2\ln 4$

$\implies 2k=2\ln 4$

$\implies k=\ln 4$

Substitute the found value of k into the formula, along with one of the points, and solve for a:
$\implies ae^{\ln 4}=40$

$\implies 4a=40$

$\implies a=10$

Therefore, the exponential function for points (1, 40) and (3, 640) is:

$\implies y=10e^{x\ln4}$

$\implies y=10(e^{\ln 4})^x$

$\implies y=10(4)^x$

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Given points from the graph:

• (-3, 10.8)
• (-2, 3.6)

Substitute the points into the formula to create two equations:

$\implies ae^{-2k}=3.6$

$\implies ae^{-3k}=10.8$

Divide the equations to eliminate a and solve for k:

$\implies \dfrac{ae^{-3k}}{ae^{-2k}}=\dfrac{10.8}{3.6}$

$\implies \dfrac{e^{-3k}}{e^{-2k}}=3$

$\implies e^{-k}=3$

$\implies \ln e^{-k}=\ln 3$

$\implies -k=\ln 3$

$\implies k=-\ln 3$

Substitute the found value of k into the formula, along with one of the points, and solve for a:
$\implies ae^{2\ln 3}=3.6$

$\implies 9a=3.6$

$\implies a=0.4$

Therefore, the exponential function for points (1, 40) and (3, 640) is:

$\implies y=0.4e^{-x\ln3}$

$\implies y=0.4(e^{\ln 3})^{-x}$

$\implies y=0.4(3)^{-x}$