Question
19 of 2019 of 20 Items
x-values y-vaules
0 5
1 35
2 245
3 1715
4 12005
5 84035
Question
Use the table to answer the question.
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)=7(5)x
and exponential function
f of x is equal to 7 times 5 to the x th power and exponential function
f(x)=5(7)x
and exponential function
f of x is equal to 5 times 7 to the x th power and exponential function
f(x)=5(7)x
and quadratic function
f of x is equal to 5 times 7 to the x th power and quadratic function
f(x)=5x+30
and linear function
Answers
GPT-4o mini
To determine the type of function that best fits the dataset provided, let's analyze the y-values in relation to the x-values.
The y-values are as follows:
- \( y(0) = 5 \)
- \( y(1) = 35 \)
- \( y(2) = 245 \)
- \( y(3) = 1715 \)
- \( y(4) = 12005 \)
- \( y(5) = 84035 \)
To check whether the function is linear, exponential, or quadratic:
1. **Linear Function**: A linear function would be of the form \( f(x) = mx + b \). If we compute the differences between consecutive y-values:
- \( 35 - 5 = 30 \)
- \( 245 - 35 = 210 \)
- \( 1715 - 245 = 1470 \)
- \( 12005 - 1715 = 10290 \)
- \( 84035 - 12005 = 72030 \)
The differences are not constant; therefore, it is not a linear function.
2. **Quadratic Function**: A quadratic function has the general form \( f(x) = ax^2 + bx + c \). To test for quadratic behavior, we would typically look at the second differences of y-values. The first differences (as computed above) are not constant, hence it is unlikely to be quadratic.
3. **Exponential Function**: An exponential function typically has the form \( f(x) = ab^x \). We can test if the ratios of consecutive y-values are constant (indicative of an exponential relationship).
- \( \frac{y(1)}{y(0)} = \frac{35}{5} = 7 \)
- \( \frac{y(2)}{y(1)} = \frac{245}{35} = 7 \)
- \( \frac{y(3)}{y(2)} = \frac{1715}{245} = 7 \)
- \( \frac{y(4)}{y(3)} = \frac{12005}{1715} = 7 \)
- \( \frac{y(5)}{y(4)} = \frac{84035}{12005} = 7 \)
All ratios are equal to 7, indicating that the dataset represents an exponential function.
Based on this analysis, the function model that represents the dataset is exponential.
From the options given, the correct answer is:
**f(x) = 5(7)^x and exponential function.**
The y-values are as follows:
- \( y(0) = 5 \)
- \( y(1) = 35 \)
- \( y(2) = 245 \)
- \( y(3) = 1715 \)
- \( y(4) = 12005 \)
- \( y(5) = 84035 \)
To check whether the function is linear, exponential, or quadratic:
1. **Linear Function**: A linear function would be of the form \( f(x) = mx + b \). If we compute the differences between consecutive y-values:
- \( 35 - 5 = 30 \)
- \( 245 - 35 = 210 \)
- \( 1715 - 245 = 1470 \)
- \( 12005 - 1715 = 10290 \)
- \( 84035 - 12005 = 72030 \)
The differences are not constant; therefore, it is not a linear function.
2. **Quadratic Function**: A quadratic function has the general form \( f(x) = ax^2 + bx + c \). To test for quadratic behavior, we would typically look at the second differences of y-values. The first differences (as computed above) are not constant, hence it is unlikely to be quadratic.
3. **Exponential Function**: An exponential function typically has the form \( f(x) = ab^x \). We can test if the ratios of consecutive y-values are constant (indicative of an exponential relationship).
- \( \frac{y(1)}{y(0)} = \frac{35}{5} = 7 \)
- \( \frac{y(2)}{y(1)} = \frac{245}{35} = 7 \)
- \( \frac{y(3)}{y(2)} = \frac{1715}{245} = 7 \)
- \( \frac{y(4)}{y(3)} = \frac{12005}{1715} = 7 \)
- \( \frac{y(5)}{y(4)} = \frac{84035}{12005} = 7 \)
All ratios are equal to 7, indicating that the dataset represents an exponential function.
Based on this analysis, the function model that represents the dataset is exponential.
From the options given, the correct answer is:
**f(x) = 5(7)^x and exponential function.**