Use the table to answer the question.

Question

x-values y-values
0 5
1 35
2 245
3 1715
4 12005
5 84035
Given the following xy
-chart, create a function model. Does this illustrate a linear function, an exponential function, or a quadratic function to represent the dataset?

(1 point)
Responses

f(x)=5(7)x
and quadratic function
f left parenthesis x right parenthesis equals 5 left parenthesis 7 right parenthesis superscript x baseline and quadratic function

f(x)=7(5)x
and exponential function
f left parenthesis x right parenthesis equals 7 left parenthesis 5 right parenthesis superscript x baseline and exponential function

f(x)=5x+30
and linear function
f left parenthesis x right parenthesis equals 5 x plus 30 and linear function

f(x)=5(7)x
and exponential function
f left parenthesis x right parenthesis equals 5 left parenthesis 7 right parenthesis superscript x baseline and exponential function
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Answer 1

To determine the best fitting model for the given dataset, we need to analyze the pattern of growth in the y-values relative to the x-values.

The given data points are:

• \( (0, 5) \)
• \( (1, 35) \)
• \( (2, 245) \)
• \( (3, 1715) \)
• \( (4, 12005) \)
• \( (5, 84035) \)

Analysis:

At first glance, we can observe that the y-values are increasing rapidly as the x-values increase. This suggests that the relationship might not be linear since linear functions grow at a constant rate.

To further analyze the growth, we can look at the ratios of consecutive y-values:

• From \( 5 \) to \( 35 \): \( \frac{35}{5} = 7 \)
• From \( 35 \) to \( 245 \): \( \frac{245}{35} = 7 \)
• From \( 245 \) to \( 1715 \): \( \frac{1715}{245} = 7 \)
• From \( 1715 \) to \( 12005 \): \( \frac{12005}{1715} = 7 \)
• From \( 12005 \) to \( 84035 \): \( \frac{84035}{12005} = 7 \)

Conclusion:

Since the ratios of consecutive y-values are constant (specifically, each value is multiplied by 7 to get the next value), this indicates that the function is indeed exponential.

The pattern can be translated into the function model:

\[ f(x) = 5 \cdot 7^x \]

This matches one of the response options:

f(x) = 5(7^x) and exponential function

So, the correct answer is:

f(x) = 5(7^x) and exponential function.