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,

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