Use the table to answer the question.

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)=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
f left parenthesis x right parenthesis equals 12 superscript x baseline
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Answer 1

To find an exponential model of the dataset presented in the table with the x-values and corresponding y-values, we want to identify the equation in the form of \( f(x) = a \cdot b^x \).

1. Look at the y-values:

   • For \( x = 0 \), \( y = 6 \).
   • For \( x = 1 \), \( y = 12 \).
   • For \( x = 2 \), \( y = 24 \).
   • For \( x = 3 \), \( y = 48 \).

2. Identify the base of the exponential model:

   • The y-values are increasing, so we will analyze the ratios:
     • \( \frac{12}{6} = 2 \) (from \( x = 0 \) to \( x = 1 \))
     • \( \frac{24}{12} = 2 \) (from \( x = 1 \) to \( x = 2 \))
     • \( \frac{48}{24} = 2 \) (from \( x = 2 \) to \( x = 3 \))

   Each time x increases by 1, the y-value doubles. This indicates that the base \( b \) is 2.

3. Use the initial value:

   • Since \( f(0) = 6 \), we can define our function starting point \( a \) as 6.

Putting it all together, we get the function:

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

Among your response options, the correct model is:

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