Question
The table of values shown below represents a linear function. Which of these points could also be an ordered pair in the table, and why?
x
0
3
6
9
12
y
3
7
11
15
19
(18, 27), because the rate of change of the function is Three-fourths
(18, 27), because the rate of change of the function is Four-thirds
(27, 18), because the rate of change of the function is Three-fourths
(27, 18), because the rate of change of the function is Four-thirds
x
0
3
6
9
12
y
3
7
11
15
19
(18, 27), because the rate of change of the function is Three-fourths
(18, 27), because the rate of change of the function is Four-thirds
(27, 18), because the rate of change of the function is Three-fourths
(27, 18), because the rate of change of the function is Four-thirds
Answers
GPT-4o mini
To determine which point could be an ordered pair in the table representing a linear function, we first need to find the rate of change (slope) between the x and y values in the provided table.
Looking at the x and y values:
- When x = 0, y = 3
- When x = 3, y = 7
- When x = 6, y = 11
- When x = 9, y = 15
- When x = 12, y = 19
We can calculate the rate of change (slope) between two points. For example, let's calculate the slope between (0, 3) and (3, 7):
\[
\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{3 - 0} = \frac{4}{3}
\]
Now, let's check the slope between (3, 7) and (6, 11):
\[
\text{Slope} = \frac{11 - 7}{6 - 3} = \frac{4}{3}
\]
The slope between the other points will yield the same result (4/3), confirming it is consistent throughout.
Next, we'll analyze the potential points:
1. **(18, 27)**: To check if this point could be on the line, we calculate if it maintains the same slope. Using the last point (12, 19):
\[
\text{Slope} = \frac{27 - 19}{18 - 12} = \frac{8}{6} = \frac{4}{3}
\]
2. **(27, 18)**: We do the same check:
\[
\text{Slope} = \frac{18 - 19}{27 - 12} = \frac{-1}{15}
\]
This does not maintain the slope of \( \frac{4}{3} \).
Since (18, 27) maintains the slope of the linear function, while (27, 18) does not, the correct statement is:
**(18, 27), because the rate of change of the function is Four-thirds.**
Looking at the x and y values:
- When x = 0, y = 3
- When x = 3, y = 7
- When x = 6, y = 11
- When x = 9, y = 15
- When x = 12, y = 19
We can calculate the rate of change (slope) between two points. For example, let's calculate the slope between (0, 3) and (3, 7):
\[
\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7 - 3}{3 - 0} = \frac{4}{3}
\]
Now, let's check the slope between (3, 7) and (6, 11):
\[
\text{Slope} = \frac{11 - 7}{6 - 3} = \frac{4}{3}
\]
The slope between the other points will yield the same result (4/3), confirming it is consistent throughout.
Next, we'll analyze the potential points:
1. **(18, 27)**: To check if this point could be on the line, we calculate if it maintains the same slope. Using the last point (12, 19):
\[
\text{Slope} = \frac{27 - 19}{18 - 12} = \frac{8}{6} = \frac{4}{3}
\]
2. **(27, 18)**: We do the same check:
\[
\text{Slope} = \frac{18 - 19}{27 - 12} = \frac{-1}{15}
\]
This does not maintain the slope of \( \frac{4}{3} \).
Since (18, 27) maintains the slope of the linear function, while (27, 18) does not, the correct statement is:
**(18, 27), because the rate of change of the function is Four-thirds.**