QuestionThe following table represents an exponential function. \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 0 & 4 \\ \hline 1 & 2 \\ \hline 2 & 1 \\ \hline 3 & $\frac{1}{2}$ \\ \hline 4 & $\frac{1}{4}$ \\ \hline \end{tabular}
The exponential function represented by the table can be written in the form $y=a b^{x}$ . Find the values for $a$ and $b$ . $a=\square$ $b=$ $\square$

Studdy Solution

STEP 1

1. The function is exponential and can be expressed as $y = a b^x$ .
2. The table provides specific values of $x$ and corresponding $y$ .

STEP 2

1. Use the initial value to determine $a$ .
2. Use the ratio of consecutive $y$ -values to determine $b$ .
3. Verify the values of $a$ and $b$ using additional points.

STEP 3

The value of $y$ when $x = 0$ is 4. Since $y = a b^0 = a$ , we have: $a = 4$

STEP 4

To find $b$ , use the ratio of consecutive $y$ -values. For example, when $x = 1$ , $y = 2$ and when $x = 0$ , $y = 4$ . Therefore: $b = \frac{y_1}{y_0} = \frac{2}{4} = \frac{1}{2}$

STEP 5

Verify the values of $a$ and $b$ using additional points. Check $x = 2$ , where $y = 1$ : $y = a b^2 = 4 \left(\frac{1}{2}\right)^2 = 4 \times \frac{1}{4} = 1$
The values are consistent with the table, confirming $a = 4$ and $b = \frac{1}{2}$ .
The exponential function is: $a = 4$ $b = \frac{1}{2}$

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