This table displays the corresponding y-values for a given set of x-values.

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)=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 for the given dataset, we can start by examining the provided y-values:

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

From the values, we can see that the y-values are increasing in a manner consistent with an exponential function.

Often, an exponential model is represented in the form \( f(x) = ab^x \), where \( a \) is the initial value when \( x = 0 \) and \( b \) is the growth factor.

Step 1: Identify \( a \)

Since the y-value when \( x = 0 \) is \( 6 \), we can say \( a = 6 \).

Step 2: Determine \( b \)

Next, we need to find the growth factor \( b \). We can use the values at \( x = 1 \) and \( x = 0 \):

• When \( x = 0 \), \( f(0) = 6 \)
• When \( x = 1 \), \( f(1) = 12 \)

Using the exponential model: \[ f(1) = ab^1 = 6b \] Setting it equal to \( 12 \): \[ 6b = 12 \implies b = \frac{12}{6} = 2 \]

Step 3: Formulate the function

Now we can write the exponential model as: \[ f(x) = 6(2^x) \]

Conclusion

The correct exponential model corresponding to the dataset shown is: \[ f(x) = 6(2^x) \]

Looking at the provided responses, the correct answer is: \[ \text{f(x) = 6(2)}^x \]