Question
On a graph the numbers on the y axis are: 10, 20 and 30. On the x axis the numbers are: 5, 10, and 15. There is a point in between 20 and 30 and the next point being on 10 on the x axis. With a negative slope. Is this proportional? What is the y intercept? What does the y intercept mean in context of this scenario? What is the slope? What does the slope mean in context of this scenario? These two points are in the first quadrant.
Answers
GPT-4o mini
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:
1. **Point A**: (10, 25)
If there is another point along the negative slope, let’s assume:
2. **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.
### 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:
1. **Point A**: (10, 25)
If there is another point along the negative slope, let’s assume:
2. **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.