Question

Use the table to answer the question.

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)=12x
f left parenthesis x right parenthesis equals 12 superscript x baseline

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

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

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

Answer 1

To find an exponential model for the given dataset, we can observe the y-values as they change with each increasing x-value. Specifically, the pattern here suggests a doubling of the y-value for each increase of 1 in x.

Let's examine the y-values:
- At \( x = 0 \), \( y = 6 \).
- When \( x = 1 \), \( y = 12 \), which is \( 6 \times 2 \).
- When \( x = 2 \), \( y = 24 \), which is \( 12 \times 2 \).
- When \( x = 3 \), \( y = 48 \), which is \( 24 \times 2 \).

It appears that from \( x = 0 \) onward, y doubles for each increment of 1 in x.

We can express this relationship in an exponential form. The general form for an exponential function is:

\[
f(x) = a \cdot b^x
\]

Where:
- \( a \) is the initial value (when \( x = 0 \)).
- \( b \) is the growth factor (how much y changes as x increases).

From observations:
- The initial value \( a \) when \( x = 0 \) is \( 6 \).
- The growth factor \( b \) is \( 2 \), since each step increases y by a factor of 2.

Thus, we can form the model:

\[
f(x) = 6(2^x)
\]

Now we can match this with the provided options:

- \( f(x) = 6(2^x) \) matches the form above.

Therefore, the correct answer is:

**f(x) = 6(2^x)**