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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

1 answer

To determine the type of function model that represents the given dataset, we can analyze the pattern in the y-values corresponding to the x-values.

The provided y-values are:

  • For x = 0, y = 5
  • For x = 1, y = 35
  • For x = 2, y = 245
  • For x = 3, y = 1715
  • For x = 4, y = 12005
  • For x = 5, y = 84035

To check if this relationship is linear, exponential, or quadratic, we can analyze the growth pattern of the y-values.

  1. Linear Function: A linear function would show a constant rate of change between the y-values as x increases. In this case, the differences between consecutive y-values are increasing rapidly.

  2. Quadratic Function: A quadratic function typically has a parabolic shape and can be represented as \(ax^2 + bx + c\). The change in differences is not constant, indicating it is not quadratic.

  3. Exponential Function: Exponential functions have a form of \(y = a \cdot b^x\) where the ratio of consecutive y-values tends to remain constant. Looking at the dataset, the growth between values seems to follow a multiplicative pattern, which suggests it might be exponential.

We can calculate the ratios of consecutive values:

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

Since the ratio of successive terms is constant indeed (approx. 7), this indicates an exponential relationship.

Given this information, we conclude that the function model can be represented as: \( f(x) = 5 \cdot 7^x \) which confirms the exponential function.

The correct response is:

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