Use the table to answer the question.

Question

x-values y-values
−1
−
1
3
0 6
1 12
2 24
3 48
Create an exponential model of the dataset shown in the xy
𝑥
𝑦
-chart.

(1 point)
Responses

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

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

f(x)=2(6)x

Answer 1

To create an exponential model from the provided dataset, we observe the pattern in the \(y\)-values as \(x\) increases.

From the given data:

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

Notice that:

• \(y\) doubles as \(x\) increases by 1, suggesting a base of 2 for the exponential function.

We can express the model as:

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

Substituting the known values to find \(a\) and \(b\):

Starting with \(f(0) = 6\):

\[ f(0) = a \cdot b^0 \Rightarrow 6 = a \cdot 1 \Rightarrow a = 6 \]

Next, we need to find \(b\). Using \(f(1) = 12\):

\[ f(1) = 6 \cdot b^1 = 12 \Rightarrow b = \frac{12}{6} = 2 \]

Now we have:

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

Thus, the exponential model that fits the data is:

\[ \boxed{f(x) = 6(2^x)} \]

From your provided options, this corresponds to:

f(x) = 6(2)^x (which is the first option given).