# Find Expontntial Model that fits the points shown in the graph of table01510102841.jpg

Question
Exponential models

Find Expontntial Model that fits the points shown in the graph of table

2020-12-18
Let us consider the following exponential model
$$\displaystyle{y}={a}{e}^{{{x}}}$$
Now, with given point (0,5), the equation becomes
$$\displaystyle{5}={a}{e}^{{{b}{\left({0}\right)}}}$$
$$\displaystyle{5}={a}{e}^{{0}}$$
Therefore,
a=5
With the given point (4,1) and a=1, the equation becomes
$$\displaystyle{1}={5}{e}^{{{b}{\left({4}\right)}}}$$
Dividing both sides by 5
$$\displaystyle\frac{{1}}{{5}}={e}^{{{4}{b}}}$$
Taking natural logarithm to the base e
$$\displaystyle\frac{{\ln{{1}}}}{{5}}={{\ln{{e}}}^{{{4}{b}}}}$$
Using the inverse property $$\displaystyle{{\ln{{e}}}^{{x}}}$$
$$\displaystyle\frac{{\ln{{1}}}}{{5}}={4}{b}$$
Therefore,
$$\displaystyle{b}=\frac{{\frac{{\ln{{1}}}}{{5}}}}{{4}}$$
b=-0.4024
Hence, the equation of the curve with a=5 and b=-0.4024 is $$\displaystyle{y}={e}^{{-{0.4024}{x}}}$$

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Figure is shown below.
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Figure is shown below.
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Figure is shown below.
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