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

Transcript text: Write an exponential function in the form $y=a b^{x}$ that goes through the points $(0,4)$ and $(9,2048)$.

Solution

Solution Steps

Step 1: Identify the form of the exponential function

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

Step 2: Use the first point to find $a$

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

Step 3: Use the second point to find $b$

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: \[ b = 2 \]

Step 4: Write the final exponential function

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