Question
4. Use the graph to match the responses. (3 points) An equation that represents this graph. The y-intercept The constant of proportionality
Look at this imaginary graph we have the y-intercept and x-intercept the pin pointed points are in the top right box of the graph theres 4 boxes but were focusing on the top right one, the pin point points are going in a diagnol line, the y line lists Miles walked as the x line lists hours, the pin point diagnol line starts at 2 4 6 8, and ends at 10
Look at this imaginary graph we have the y-intercept and x-intercept the pin pointed points are in the top right box of the graph theres 4 boxes but were focusing on the top right one, the pin point points are going in a diagnol line, the y line lists Miles walked as the x line lists hours, the pin point diagnol line starts at 2 4 6 8, and ends at 10
Answers
GPT-4o mini
To match the responses based on the description of the graph you provided, let's address each of the components:
1. **An equation that represents this graph**: If the graph depicts a diagonal line starting from the y-axis, we can determine the slope (rate of change) and the y-intercept. Given the coordinates you mentioned (4, 2), (6, 4), (8, 6), and (10, 8), the relationship appears to be linear, and can be represented as \(y = mx + b\).
To find the slope (\(m\)):
\[
m = \frac{\Delta y}{\Delta x} = \frac{4 - 2}{6 - 4} = 1
\]
Therefore, the equation becomes:
\[
y = x - 2
\]
2. **The y-intercept**: The y-intercept is the value of \(y\) when \(x = 0\). From the given points, it appears that the line intersects the y-axis at \(y = -2\). Thus, the y-intercept is:
\[
-2
\]
3. **The constant of proportionality**: In the context of slope/linear equations, the constant of proportionality refers to the slope \(m\). Here, from the earlier calculation, we found that the slope is 1. So, the constant of proportionality is:
\[
1
\]
To summarize, based on your graph:
- The equation that represents the graph is \(y = x - 2\).
- The y-intercept is \(-2\).
- The constant of proportionality is \(1\).
1. **An equation that represents this graph**: If the graph depicts a diagonal line starting from the y-axis, we can determine the slope (rate of change) and the y-intercept. Given the coordinates you mentioned (4, 2), (6, 4), (8, 6), and (10, 8), the relationship appears to be linear, and can be represented as \(y = mx + b\).
To find the slope (\(m\)):
\[
m = \frac{\Delta y}{\Delta x} = \frac{4 - 2}{6 - 4} = 1
\]
Therefore, the equation becomes:
\[
y = x - 2
\]
2. **The y-intercept**: The y-intercept is the value of \(y\) when \(x = 0\). From the given points, it appears that the line intersects the y-axis at \(y = -2\). Thus, the y-intercept is:
\[
-2
\]
3. **The constant of proportionality**: In the context of slope/linear equations, the constant of proportionality refers to the slope \(m\). Here, from the earlier calculation, we found that the slope is 1. So, the constant of proportionality is:
\[
1
\]
To summarize, based on your graph:
- The equation that represents the graph is \(y = x - 2\).
- The y-intercept is \(-2\).
- The constant of proportionality is \(1\).
this imaginary line starts at the origin going diagonally to the right going straight
GPT-4o mini
Thank you for the clarification! If the imaginary line starts at the origin (0,0) and goes diagonally to the right, it indicates a direct proportional relationship between the two variables: hours and miles walked.
Here’s how we can summarize the match:
1. **An equation that represents this graph**: Since the line starts at the origin and increases diagonally, the equation is typically of the form:
\[
y = mx
\]
If the points mentioned (2, 4), (4, 8), etc., suggest that for every 2 hours, 4 miles are walked, it means the slope (constant of proportionality) is 2. Therefore, the equation can be written as:
\[
y = 2x
\]
2. **The y-intercept**: The y-intercept is the point where the line crosses the y-axis. Since the line starts at the origin (0,0), the y-intercept is:
\[
0
\]
3. **The constant of proportionality**: The constant of proportionality is equal to the slope of the line. In this case, the constant of proportionality (slope) is:
\[
2
\]
To summarize, based on your description of the imaginary graph:
- The equation that represents the graph is \(y = 2x\).
- The y-intercept is \(0\).
- The constant of proportionality is \(2\).
Here’s how we can summarize the match:
1. **An equation that represents this graph**: Since the line starts at the origin and increases diagonally, the equation is typically of the form:
\[
y = mx
\]
If the points mentioned (2, 4), (4, 8), etc., suggest that for every 2 hours, 4 miles are walked, it means the slope (constant of proportionality) is 2. Therefore, the equation can be written as:
\[
y = 2x
\]
2. **The y-intercept**: The y-intercept is the point where the line crosses the y-axis. Since the line starts at the origin (0,0), the y-intercept is:
\[
0
\]
3. **The constant of proportionality**: The constant of proportionality is equal to the slope of the line. In this case, the constant of proportionality (slope) is:
\[
2
\]
To summarize, based on your description of the imaginary graph:
- The equation that represents the graph is \(y = 2x\).
- The y-intercept is \(0\).
- The constant of proportionality is \(2\).
so is that like (0,0) and (0,2) (2,0) or just 0 and 2
GPT-4o mini
Let's clarify the terminology and the points based on your earlier description of the graph.
1. **(0,0)**: This point is where the line starts, known as the origin. Here, when \(x = 0\) (0 hours), \(y = 0\) (0 miles walked).
2. **The y-intercept**: This is identified by the point where the line crosses the y-axis. In your case, since the line starts at the origin, when \(x = 0\), the corresponding \(y\) value is also 0. So the y-intercept is at the point **(0, 0)**.
3. **Constant of proportionality**: If the equation representing the relationship is \(y = mx\), where \(m\) is the constant of proportionality, in your example it appears to be \(2\), which means for every hour walked, if \(m\) is 2, the equation would be \(y = 2x\). This means for every hour, the distance in miles increases by 2.
4. **Key points**:
- If you plotted this on a graph, some example points would be:
- (0, 0) when 0 hours are walked, 0 miles are walked.
- (1, 2) when 1 hour is walked, 2 miles are walked.
- (2, 4) when 2 hours are walked, 4 miles are walked.
- (3, 6) when 3 hours are walked, 6 miles are walked.
To summarize:
The important points based on your equation \(y = 2x\) would specifically be:
- The y-intercept is **0** (this point is (0,0)).
- The constant of proportionality is **2**.
So, the points you're asking about like (0,2) or (2,0) would not apply here, as in the context we discussed the key point on the y-axis is simply **(0,0)**, with the line defined by the equation representing the relationship between hours and miles walked.
1. **(0,0)**: This point is where the line starts, known as the origin. Here, when \(x = 0\) (0 hours), \(y = 0\) (0 miles walked).
2. **The y-intercept**: This is identified by the point where the line crosses the y-axis. In your case, since the line starts at the origin, when \(x = 0\), the corresponding \(y\) value is also 0. So the y-intercept is at the point **(0, 0)**.
3. **Constant of proportionality**: If the equation representing the relationship is \(y = mx\), where \(m\) is the constant of proportionality, in your example it appears to be \(2\), which means for every hour walked, if \(m\) is 2, the equation would be \(y = 2x\). This means for every hour, the distance in miles increases by 2.
4. **Key points**:
- If you plotted this on a graph, some example points would be:
- (0, 0) when 0 hours are walked, 0 miles are walked.
- (1, 2) when 1 hour is walked, 2 miles are walked.
- (2, 4) when 2 hours are walked, 4 miles are walked.
- (3, 6) when 3 hours are walked, 6 miles are walked.
To summarize:
The important points based on your equation \(y = 2x\) would specifically be:
- The y-intercept is **0** (this point is (0,0)).
- The constant of proportionality is **2**.
So, the points you're asking about like (0,2) or (2,0) would not apply here, as in the context we discussed the key point on the y-axis is simply **(0,0)**, with the line defined by the equation representing the relationship between hours and miles walked.
so do I place 0,0 or 0 there
GPT-4o mini
In the context of identifying and placing values based on your graph:
1. **For the y-intercept**: You would put **0**. This indicates the y-value where the line crosses the y-axis, which is at the point (0, 0).
2. **For the point itself**: If you were to list the coordinate of the y-intercept, it would be (0, 0).
So if you need to specify just the y-intercept, you would place **0**. If you need to specify the coordinate as a point on the graph, you would write it as **(0, 0)**.
### Summary:
- If asked for just the y-intercept: **0**
- If asked for the y-intercept as a coordinate: **(0, 0)**
1. **For the y-intercept**: You would put **0**. This indicates the y-value where the line crosses the y-axis, which is at the point (0, 0).
2. **For the point itself**: If you were to list the coordinate of the y-intercept, it would be (0, 0).
So if you need to specify just the y-intercept, you would place **0**. If you need to specify the coordinate as a point on the graph, you would write it as **(0, 0)**.
### Summary:
- If asked for just the y-intercept: **0**
- If asked for the y-intercept as a coordinate: **(0, 0)**