Question

Use the table to answer the question.

This table displays the corresponding y-values for a given set of x-values.
x-values y-values
−1
3
0 6
1 12
2 24
3 48
Create an exponential model of the dataset shown in the xy
-chart.




f(x)=6(2)x
f left parenthesis x right parenthesis equals 6 left parenthesis 2 right parenthesis superscript x baseline

f(x)=6(2x)
f left parenthesis x right parenthesis equals 6 left parenthesis 2 x right parenthesis

f(x)=2(6)x
f left parenthesis x right parenthesis equals 2 left parenthesis 6 right parenthesis superscript x baseline

f(x)=12x

Answer 1

To create an exponential model based on the given dataset:

1. **Identify the Pattern**: First, let's observe the \( y \)-values:
- When \( x = 0 \), \( y = 6 \)
- When \( x = 1 \), \( y = 12 \)
- When \( x = 2 \), \( y = 24 \)
- When \( x = 3 \), \( y = 48 \)

We can see that as \( x \) increases by 1, the \( y \)-values are doubling. This suggests that the function could be of the form \( f(x) = a \cdot b^x \).

2. **Determine Initial Value**: The value of \( y \) when \( x = 0 \) provides the initial value. Here, \( a = 6 \).

3. **Determine the Growth Factor**: From \( x = 0 \) to \( x = 1 \):
- \( \frac{y(1)}{y(0)} = \frac{12}{6} = 2 \)
This indicates that the base \( b \) of our exponential function is 2.

4. **Construct the Model**: Substituting \( a \) and \( b \), the exponential function can be written as:
\[
f(x) = 6 \cdot 2^x
\]

Based on the choices given in the original question:

- The correct exponential model is:
\[
f(x) = 6(2^x)
\]

Thus, the answer is:
\[
f(x) = 6(2^x)
\]