Questions: Find the equation of the exponential function represented by the table below: x y 0 2 1 8 2 32 3 128

Transcript text: Find the equation of the exponential function represented by the table below: \begin{tabular}{|c|c|} \hline$x$ & $y$ \\ \hline 0 & 2 \\ \hline 1 & 8 \\ \hline 2 & 32 \\ \hline 3 & 128 \\ \hline \end{tabular}

Solution

Solution Steps

Step 1: Identify the general form of the exponential function

An exponential function can be written in the form: \[ y = a \cdot b^x \] where:

$a$ is the initial value (when $x = 0$),
$b$ is the base (growth factor).

Step 2: Determine the initial value $a$

From the table, when $x = 0$, $y = 2$. Substituting into the general form: \[ 2 = a \cdot b^0 \] Since $b^0 = 1$, this simplifies to: \[ a = 2 \]

Step 3: Determine the base $b$

Using the value of $a = 2$ and the next point from the table ($x = 1$, $y = 8$): \[ 8 = 2 \cdot b^1 \] Solving for $b$: \[ b = \frac{8}{2} = 4 \]

Step 4: Write the equation of the exponential function

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

Final Answer

The equation of the exponential function is: \[ \boxed{y = 2 \cdot 4^x} \]

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