To analyze whether the relationship represented by the points on the graph is proportional and to derive additional information such as the y-intercept and slope, let’s break down the details provided.
Proportionality
A relationship is proportional if the ratio of \( y \) to \( x \) is constant. Given that the points are in the first quadrant, and you describe a point in between 20 and 30 on the y-axis when aligned with 10 on the x-axis, let’s assume that the specific point you are referring to is around (10, 25) to illustrate:
- Point A: (10, 25)
If there is another point along the negative slope, let’s assume:
- Point B: (5, 30) (for the sake of example)
To check for proportionality, we can calculate the ratio of \( y \) to \( x \) for both points:
- For Point A: \( \frac{y}{x} = \frac{25}{10} = 2.5 \)
- For Point B: \( \frac{y}{x} = \frac{30}{5} = 6 \)
Since these ratios are not equal, the relationship is not proportional.
Y-Intercept
To find the y-intercept of a line, we typically express the equation of the line in slope-intercept form, \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
To find this equation, we first determine the slope \( m \):
- Using the two points (let’s say (10, 25) and (5, 30)):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{30 - 25}{5 - 10} = \frac{5}{-5} = -1 \]
So, the slope \( m \) is -1.
Now, using the point-slope form \( y - y_1 = m(x - x_1) \):
Using point (10, 25):
\[ y - 25 = -1(x - 10) \]
Expanding this:
\[ y - 25 = -x + 10 \]
Therefore,
\[ y = -x + 35 \]
From this equation, we see that the y-intercept (\( b \)) is 35 (the point where the line crosses the y-axis).
Meaning of Y-Intercept
In the context of this scenario, the y-intercept (35) means that when there is no input from the x-axis (i.e., \( x = 0 \)), the output value on the y-axis would be 35. This might represent a starting value or initial condition prior to any change.
Slope
The slope \( m = -1 \) indicates how much \( y \) changes for a unit change in \( x \):
Meaning of Slope
In context, a slope of -1 signifies that for every increase of 1 unit along the x-axis, the value of y decreases by 1 unit. This suggests an inverse relationship between x and y; as \( x \) increases, \( y \) decreases.
Summary
- The relationship is not proportional.
- The y-intercept is 35, indicating the value of y when x equals 0.
- The slope is -1, indicating an inverse relationship between the x- and y-values.