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)=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

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
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Answer 1

To determine the type of function that fits the given data points, we should analyze the patterns in the y-values corresponding to the x-values.

The provided data is:

x-values | y-values
--------------------
0        | 5
1        | 35
2        | 245
3        | 1715
4        | 12005
5        | 84035

1. Calculate the ratios of successive y-values:
   • 35 / 5 = 7
   • 245 / 35 = 7
   • 1715 / 245 = 7
   • 12005 / 1715 = 7
   • 84035 / 12005 = 7

The ratios are consistent; hence it suggests that the relationship is exponential.

2. The general form of an exponential function is: \[ f(x) = a \cdot b^x \] From the calculations, we can see that the y-values are increasing by a factor of 7 with each increase in x.

3. Identifying \( a \) and \( b \): Here:

   • \( a = 5 \) (the value when \( x = 0 \) is \( 5 = 5 \cdot 7^0 \))
   • \( b = 7 \)

Thus, the function that models the dataset is: \[ f(x) = 5 \cdot 7^x \]

Conclusion: This illustrates an exponential function.

Therefore, the correct response is: f(x) = 5(7)^x and exponential function.