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Use the table to answer the 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)=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)=7(5)x
and exponential function

1 answer

To determine the type of function that best fits the dataset provided, we can analyze the progression of the y-values based on the x-values.

Here are the y-values:

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

As we analyze the pattern:

  1. Check for Linear: For a linear function, the differences between consecutive y-values should be constant. Calculating these differences shows they are not constant (35 - 5, 245 - 35, etc.), indicating that this is not a linear function.

  2. Check for Quadratic: For a quadratic function, the second differences of y-values (the differences of the differences) should be constant. However, due to the rapid increase in y-values (the growth appears to be multiplicative rather than additive), it is likely not quadratic as well.

  3. Check for Exponential: For an exponential function, the ratio of consecutive y-values should be approximately constant. Calculating these ratios:

  • \( 35 / 5 = 7 \)
  • \( 245 / 35 = 7 \)
  • \( 1715 / 245 = 7 \)
  • \( 12005 / 1715 = 7 \)
  • \( 84035 / 12005 = 7 \)

Since the ratios are consistently 7, this indicates that the function fits the form \( f(x) = a \cdot b^x \), where \( a = 5 \) and \( b = 7 \). Therefore, the model can be expressed as:

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

Based on this analysis, the correct response is:

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