Question
The table below represents a linear function. Identify the rate of change of the function.
xx yy
44 88
66 1111
88 1414
1010 1717
xx yy
44 88
66 1111
88 1414
1010 1717
Answers
GPT-4o mini
To find the rate of change of the linear function represented by the given table, we can calculate the change in \( y \) over the change in \( x \) between two points.
The formula for the rate of change (slope) is given by:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's select the first two points in the table:
- Point 1: \( (x_1, y_1) = (44, 88) \)
- Point 2: \( (x_2, y_2) = (66, 1111) \)
Now we can substitute these values into the formula:
\[
\text{slope} = \frac{1111 - 88}{66 - 44}
\]
\[
= \frac{1023}{22}
\]
\[
= 46.47727272727273
\]
Now, let's check the other points to ensure consistency in the slope:
Using points \( (66, 1111) \) and \( (88, 1414) \):
\[
\text{slope} = \frac{1414 - 1111}{88 - 66} = \frac{303}{22} = 13.772727272727273
\]
Using points \( (88, 1414) \) and \( (1010, 1717) \):
\[
\text{slope} = \frac{1717 - 1414}{1010 - 88} = \frac{303}{922} \approx 0.328
\]
Calculating the slopes resulted in varying values, which indicates that this relationship does not maintain a consistent linear function across all given data points. Hence, it does not represent a linear function effectively.
From the calculated slopes, it’s evident that the values don't establish a constant rate of change, indicating that more checks are needed or an alternative representation approach to further analyze the function might be beneficial if treated linearly.
The formula for the rate of change (slope) is given by:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's select the first two points in the table:
- Point 1: \( (x_1, y_1) = (44, 88) \)
- Point 2: \( (x_2, y_2) = (66, 1111) \)
Now we can substitute these values into the formula:
\[
\text{slope} = \frac{1111 - 88}{66 - 44}
\]
\[
= \frac{1023}{22}
\]
\[
= 46.47727272727273
\]
Now, let's check the other points to ensure consistency in the slope:
Using points \( (66, 1111) \) and \( (88, 1414) \):
\[
\text{slope} = \frac{1414 - 1111}{88 - 66} = \frac{303}{22} = 13.772727272727273
\]
Using points \( (88, 1414) \) and \( (1010, 1717) \):
\[
\text{slope} = \frac{1717 - 1414}{1010 - 88} = \frac{303}{922} \approx 0.328
\]
Calculating the slopes resulted in varying values, which indicates that this relationship does not maintain a consistent linear function across all given data points. Hence, it does not represent a linear function effectively.
From the calculated slopes, it’s evident that the values don't establish a constant rate of change, indicating that more checks are needed or an alternative representation approach to further analyze the function might be beneficial if treated linearly.