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.

(1 point)
Responses

f(x)=2(6)x
f left parenthesis x right parenthesis equals 2 left parenthesis 6 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)=12x
f left parenthesis x right parenthesis equals 12 superscript x baseline

f(x)=6(2)x

Answer 1

To find an exponential model for the dataset provided, we first need to analyze the y-values corresponding to the x-values.

From the table:
- When \( x = 0 \), \( y = 6 \).
- When \( x = 1 \), \( y = 12 \).
- When \( x = 2 \), \( y = 24 \).
- When \( x = 3 \), \( y = 48 \).

Let's look for a pattern in the y-values:

- The ratio of \( y \)-values when \( x \) increases by 1:
- From \( 0 \) to \( 1 \): \( \frac{12}{6} = 2 \)
- From \( 1 \) to \( 2 \): \( \frac{24}{12} = 2 \)
- From \( 2 \) to \( 3 \): \( \frac{48}{24} = 2 \)

This indicates that the y-values double as x increases by 1, suggesting an exponential growth with a base of 2.

The general form of an exponential function can be expressed as:
\[
f(x) = a \cdot b^x
\]
where \( a \) is the initial value (y-intercept when \( x = 0 \)) and \( b \) is the growth factor.

From the table, when \( x = 0 \), \( y = 6 \), giving us \( a = 6 \) and since we found that \( b = 2 \), we can write the model as:
\[
f(x) = 6 \cdot 2^x
\]

Now, let's check our options:
1. \( f(x) = 2(6)^x \)
2. \( f(x) = 6(2^x) \)
3. \( f(x) = 12^x \)
4. \( f(x) = 6(2)^x \)

The correct exponential model for the data provided is:
\[
f(x) = 6(2^x)
\]
Thus, the answer is:
**f(x) = 6(2^x)**