Questions: Write an exponential function in the form y=ab^x that goes through the points (0,4) and (9,2048).

Solution Steps

We need to find an exponential function in the form $y = ab^x$ that passes through the points $(0, 4)$ and $(9, 2048)$.

Substitute the point $(0, 4)$ into the equation $y = ab^x$: $$4 = ab^0$$ Since $b^0 = 1$, we have: $$a = 4$$

Substitute the point $(9, 2048)$ and $a = 4$ into the equation $y = ab^x$: $$2048 = 4b^9$$ Divide both sides by 4: $$512 = b^9$$ Solve for $b$ by taking the ninth root of both sides: $$b = \sqrt[9]{512}$$ Since $512 = 2^9$, we have: $$b = 2$$

Substitute $a = 4$ and $b = 2$ back into the form $y = ab^x$: $$y = 4 \cdot 2^x$$

Final Answer