To determine the exponential function that fits the given set of values, we follow these steps:
1. Identify the Pattern in the Table:
We notice that the [tex]\( y \)[/tex]-values in the table increase exponentially as [tex]\( x \)[/tex] increases.
2. Calculate the Ratios of Consecutive [tex]\( y \)[/tex]-Values:
To find the base [tex]\( b \)[/tex] of the exponential function [tex]\( f(x) = a \cdot b^x \)[/tex], we compute the ratios of consecutive [tex]\( y \)[/tex]-values:
- For [tex]\( x = 0 \)[/tex] and [tex]\( x = 1 \)[/tex]:
[tex]\[
\text{Ratio}_1 = \frac{y_1}{y_0} = \frac{10.5}{3.5} = 3.0
\][/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( x = 2 \)[/tex]:
[tex]\[
\text{Ratio}_2 = \frac{y_2}{y_1} = \frac{31.5}{10.5} = 3.0
\][/tex]
- For [tex]\( x = 2 \)[/tex] and [tex]\( x = 3 \)[/tex]:
[tex]\[
\text{Ratio}_3 = \frac{y_3}{y_2} = \frac{94.5}{31.5} = 3.0
\][/tex]
Since all these ratios are equal and constant, we confirm that the common ratio (base of the exponential function) is [tex]\( b = 3 \)[/tex].
3. Determine the Initial Value [tex]\( a \)[/tex]:
The initial value [tex]\( a \)[/tex] of the function is the value of [tex]\( y \)[/tex] when [tex]\( x = 0 \)[/tex]. From the table, we see that when [tex]\( x = 0 \)[/tex], [tex]\( y = 3.5 \)[/tex]. Thus, [tex]\( a = 3.5 \)[/tex].
4. Formulate the Exponential Function:
Now that we have [tex]\( a = 3.5 \)[/tex] and [tex]\( b = 3 \)[/tex], we can write the function as:
[tex]\[
f(x) = a \cdot b^x = 3.5 \cdot 3^x
\][/tex]
5. Select the Correct Answer:
The given options are:
- A. [tex]\( f(x) = 3(3.5)^x \)[/tex]
- B. [tex]\( f(x) = 3.5(3)^x \)[/tex]
- C. [tex]\( f(x) = 10.5(3)^x \)[/tex]
- D. [tex]\( f(x) = 3.5\left(\frac{1}{3}\right)^x \)[/tex]
The correct exponential function that matches our findings is:
[tex]\[
f(x) = 3.5 \cdot 3^x
\][/tex]
Thus, the correct answer is:
B. [tex]\( f(x) = 3.5(3)^x \)[/tex].
