Which equation is the most appropriate exponential model for the data shown in the scatterplot?

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Exponential functions can be modeled using the equation,

$$y=a(b)^x$$

$a=$ initial value, or $y$-intercept

$b=$ multiplier or growth factor

From the graph we can estimate the $y$-intercept and multipier. From the answer choices, the most reasonable $y$-intercept would be $(0,10)$. Every time the $x$ value is increased by 1, the $y$ value approximately doubles, giving us a multiplier of 2.

With initial value 10 and multiplier 2,

$$y=a(b)^x$$ $$\boxed{y=10(2)^x}$$

We can derive an approximate equation by choosing points from the scatterplot. Using an approximate $y$-intercept $(0,10)$ to find $a$.

$$y=a(b)^x$$ $$10=a(b)^0$$ $$10=a(1)$$ $$10=a$$

Using another point $(3,80)$ to find $b$,

$$y=a(b)^x$$ $$80=10(b)^3$$ $$8=b^3$$ $$\sqrt[3]{8}=b$$ $$2=b$$

Substituting back in,

$$y=a(b)^x$$ $$\boxed{y=10(2)^x}$$