Question
Use the graph to match the responses.
2 miles walked is 1 hour, 4 miles is 2, 6 miles walked is 3 and so on
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The y-intercept
The constant of proportionality
An equation that represents this graph.
2 miles walked is 1 hour, 4 miles is 2, 6 miles walked is 3 and so on
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The y-intercept
The constant of proportionality
An equation that represents this graph.
Answers
GPT-4o mini
Based on the information provided about the relationship between miles walked and time, we can analyze the various components requested.
1. **The y-intercept**: This represents the point where the line crosses the y-axis. In this case, when 0 miles are walked, the time is 0 hours. Thus, the y-intercept is (0,0).
2. **The constant of proportionality**: The constant of proportionality is the rate of change from one variable to another. Here, for every 2 miles walked, it takes 1 hour. Therefore, the constant of proportionality (rate of miles per hour) is 2 miles/hour.
3. **An equation that represents this graph**: The general form of the equation representing a direct relationship between miles (x) and hours (y) can be expressed as:
\[
y = \frac{1}{2}x
\]
or equivalently,
\[
y = 0.5x
\]
This indicates that for every 2 miles (x), the time (y) increases by 1 hour.
To summarize:
- **The y-intercept**: (0, 0)
- **The constant of proportionality**: 2 miles/hour
- **An equation that represents this graph**: \( y = \frac{1}{2}x \) or \( y = 0.5x \)
1. **The y-intercept**: This represents the point where the line crosses the y-axis. In this case, when 0 miles are walked, the time is 0 hours. Thus, the y-intercept is (0,0).
2. **The constant of proportionality**: The constant of proportionality is the rate of change from one variable to another. Here, for every 2 miles walked, it takes 1 hour. Therefore, the constant of proportionality (rate of miles per hour) is 2 miles/hour.
3. **An equation that represents this graph**: The general form of the equation representing a direct relationship between miles (x) and hours (y) can be expressed as:
\[
y = \frac{1}{2}x
\]
or equivalently,
\[
y = 0.5x
\]
This indicates that for every 2 miles (x), the time (y) increases by 1 hour.
To summarize:
- **The y-intercept**: (0, 0)
- **The constant of proportionality**: 2 miles/hour
- **An equation that represents this graph**: \( y = \frac{1}{2}x \) or \( y = 0.5x \)
the options: (0, 2), y = x, 1/2, (2, 0), y = 1/2x, 0, 2, (0, 0), 1, (0, 1), y = 2x
GPT-4o mini
Given the options you've provided, here's how to match them to the requested responses:
1. **The y-intercept**:
- The correct response is **(0, 0)**, as this is the point where the graph crosses the y-axis.
2. **The constant of proportionality**:
- The correct response is **1/2**. This indicates that for every 1 hour (the time), you travel 2 miles (the distance), which can be understood as a rate of 1/2 hours per mile.
3. **An equation that represents this graph**:
- The correct response is **y = 1/2x**. This equation indicates the relationship between the time (y) and the distance (x), where for every 2 miles, it takes an hour.
To summarize the matches:
- **The y-intercept**: (0, 0)
- **The constant of proportionality**: 1/2
- **An equation that represents this graph**: y = 1/2x
1. **The y-intercept**:
- The correct response is **(0, 0)**, as this is the point where the graph crosses the y-axis.
2. **The constant of proportionality**:
- The correct response is **1/2**. This indicates that for every 1 hour (the time), you travel 2 miles (the distance), which can be understood as a rate of 1/2 hours per mile.
3. **An equation that represents this graph**:
- The correct response is **y = 1/2x**. This equation indicates the relationship between the time (y) and the distance (x), where for every 2 miles, it takes an hour.
To summarize the matches:
- **The y-intercept**: (0, 0)
- **The constant of proportionality**: 1/2
- **An equation that represents this graph**: y = 1/2x
The equation of this graph is y = 2/3x − 4
What transformation would need to happen for the equation of the line to be y = 2/3x ?
(1 point)
Responses
Shift the graph right 3, down 2
Shift the graph right 3, down 2
Shift the graph up 4
Shift the graph up 4
Shift the graph down 3 and left 2
Shift the graph down 3 and left 2
Shift the graph left 4
What transformation would need to happen for the equation of the line to be y = 2/3x ?
(1 point)
Responses
Shift the graph right 3, down 2
Shift the graph right 3, down 2
Shift the graph up 4
Shift the graph up 4
Shift the graph down 3 and left 2
Shift the graph down 3 and left 2
Shift the graph left 4
GPT-4o mini
To transform the equation \( y = \frac{2}{3}x - 4 \) into the equation \( y = \frac{2}{3}x \), we need to eliminate the constant term \(-4\).
This means we need to shift the graph up by 4 units.
So, the correct response is:
**Shift the graph up 4.**
This means we need to shift the graph up by 4 units.
So, the correct response is:
**Shift the graph up 4.**
A cab company charges $8 per mile for a lift to the airport.
What change would the company make to their charges to make this a non proportional situation?
(1 point)
Responses
Charge a flat rate of $10 and then $8 per mile
Charge a flat rate of $10 and then $8 per mile
Charge $12 per mile instead of $8
Charge $12 per mile instead of $8
Charge $4 per mile instead of $8
Charge $4 per mile instead of $8
No changes are needed
No changes are needed
Question 6
7. One business charges $12 per t-shirt custom made.
Another business charges $11 per t-shirt plus a $20 set up fee for custom made.
If you were to graph both of these scenarios on the coordinate plane, the first graph would be proportional and the second non proportional.
How would you transform the first graph to make it become a non proportional graph with the same y-intercept as the second graph?
(1 point)
Responses
Right 1
Right 1
Left 1
Left 1
Left 20
Left 20
Up 1
Up 1
Right 20
Right 20
Up 20
What change would the company make to their charges to make this a non proportional situation?
(1 point)
Responses
Charge a flat rate of $10 and then $8 per mile
Charge a flat rate of $10 and then $8 per mile
Charge $12 per mile instead of $8
Charge $12 per mile instead of $8
Charge $4 per mile instead of $8
Charge $4 per mile instead of $8
No changes are needed
No changes are needed
Question 6
7. One business charges $12 per t-shirt custom made.
Another business charges $11 per t-shirt plus a $20 set up fee for custom made.
If you were to graph both of these scenarios on the coordinate plane, the first graph would be proportional and the second non proportional.
How would you transform the first graph to make it become a non proportional graph with the same y-intercept as the second graph?
(1 point)
Responses
Right 1
Right 1
Left 1
Left 1
Left 20
Left 20
Up 1
Up 1
Right 20
Right 20
Up 20
GPT-4o mini
### Question 5
For the cab company's charging structure:
Currently, they charge $8 per mile, which is a proportional relationship (the cost is directly related to miles driven). To make this a non-proportional situation, they could charge a flat fee in addition to the per-mile rate.
**The correct response is:**
**Charge a flat rate of $10 and then $8 per mile.**
### Question 6
For the t-shirt businesses:
The first business charges $12 per t-shirt, which is a proportional relationship since there is no additional fee involved. The second business charges $11 per t-shirt plus a $20 set-up fee, making it non-proportional due to the fixed fee.
To transform the first graph (the proportional graph) so that it becomes a non-proportional graph with the same y-intercept as the second graph, you would need to introduce a fixed cost factor like the $20 set-up fee.
Since the first graph currently intercepts the y-axis at $12 (the cost of one t-shirt), you would need to account for the distance of $20–$12 = $8, by shifting the entire graph down by $20.
**The correct response is:**
**Down 20.**
However, the options given do not include “Down 20.” Instead, the best choice given that matches the intention of adjusting for that set-up fee is **Left 20**, which suggests needing to adjust the relationship to align with the fixed costs of another graph. So it seems a mistake in the interpretation of vertical shifts may exist. If based purely on available options to correlate changes:
**Choose: Up 20** (if we can assume a vertical adjustment to hit that comparison; however based on pure fee to flat costs "Left 20" as said would mean introducing an offset on cheaper side).
So if strictly shifting based on fixed comparison should lead to lower per item base cost charge, but if needing fixed correlate "Left 20" might allude to rate charge scenario mapping.
For the cab company's charging structure:
Currently, they charge $8 per mile, which is a proportional relationship (the cost is directly related to miles driven). To make this a non-proportional situation, they could charge a flat fee in addition to the per-mile rate.
**The correct response is:**
**Charge a flat rate of $10 and then $8 per mile.**
### Question 6
For the t-shirt businesses:
The first business charges $12 per t-shirt, which is a proportional relationship since there is no additional fee involved. The second business charges $11 per t-shirt plus a $20 set-up fee, making it non-proportional due to the fixed fee.
To transform the first graph (the proportional graph) so that it becomes a non-proportional graph with the same y-intercept as the second graph, you would need to introduce a fixed cost factor like the $20 set-up fee.
Since the first graph currently intercepts the y-axis at $12 (the cost of one t-shirt), you would need to account for the distance of $20–$12 = $8, by shifting the entire graph down by $20.
**The correct response is:**
**Down 20.**
However, the options given do not include “Down 20.” Instead, the best choice given that matches the intention of adjusting for that set-up fee is **Left 20**, which suggests needing to adjust the relationship to align with the fixed costs of another graph. So it seems a mistake in the interpretation of vertical shifts may exist. If based purely on available options to correlate changes:
**Choose: Up 20** (if we can assume a vertical adjustment to hit that comparison; however based on pure fee to flat costs "Left 20" as said would mean introducing an offset on cheaper side).
So if strictly shifting based on fixed comparison should lead to lower per item base cost charge, but if needing fixed correlate "Left 20" might allude to rate charge scenario mapping.
The tables below show the number of jumping jacks completed after a given period of time in minutes.
Kimberly
Time (min) # Jumping Jacks
3 28
8 58
12 82
16 106
Katrina
Time (min) # Jumping Jacks
2 16
5 40
12 96
20 160
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The person that did the most jumping jacks initially before the time started
The person showing a proportional relationship between time and # of jumping jacks
The person doing the most jumping jacks per minute
Kimberly
Time (min) # Jumping Jacks
3 28
8 58
12 82
16 106
Katrina
Time (min) # Jumping Jacks
2 16
5 40
12 96
20 160
(3 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The person that did the most jumping jacks initially before the time started
The person showing a proportional relationship between time and # of jumping jacks
The person doing the most jumping jacks per minute
GPT-4o mini
To analyze the data provided in the tables for Kimberly and Katrina, we will derive the information needed for each response:
1. **The person that did the most jumping jacks initially before the time started**:
- The initial value (when time is 0 minutes) isn't given directly in either table. However, we can infer it based on their patterns. Since both have made a baseline of jumping jacks based on their starting minutes, to derive any insights accurately we consider the starting point max leading into the interpretations afterward.
- But as far as we draw conclusions **neither has a direct initial value at t=0** from provided tables. (If any movement was extrapolated, assume it's **0 for both** unless specified otherwise as base jumps).
2. **The person showing a proportional relationship between time and the number of jumping jacks**:
- To determine proportionality, we would look at the ratio of jumping jacks to time between the moments presented.
- **Kimberly's data**: It is not linear as it varies in jumps for increments.
- **Katrina's data**: Appears to show a consistent trend; 16 jumps over 2 mins equals 8 per min; 40 over 5 mins remains aligned; repeating analysis divides through confirming proportional jumps of 8/min.
- Thus, **Katrina** is more proportional in nature.
3. **The person doing the most jumping jacks per minute**:
- Calculate the jumps per minute:
- **Kimberly**:
- 3 min: \( \frac{28}{3} \) ≈ 9.33
- 8 min: \( \frac{58}{8} \) = 7.25
- 12 min: \( \frac{82}{12} \) ≈ 6.83
- 16 min: \( \frac{106}{16} \) = 6.625
- **Katrina**:
- 2 min: \( \frac{16}{2} \) = 8
- 5 min: \( \frac{40}{5} \) = 8
- 12 min: \( \frac{96}{12} \) = 8
- 20 min: \( \frac{160}{20} \) = 8
- The jumps per minute for Kimberly declined, with highest remaining at roughly 9.33, while Katrina maintained a consistent rate of 8.
- Therefore, **Kimberly** is performing at the highest single-minute rate based on the respective highest value among entries.
Based on these analyses, the responses are:
- **The person that did the most jumping jacks initially before the time started**: **Neither (or assume both start at 0 without specified initial counts)**.
- **The person showing a proportional relationship between time and # of jumping jacks**: **Katrina**.
- **The person doing the most jumping jacks per minute**: **Kimberly**.
You can navigate and place these responses in their respective areas as needed!
1. **The person that did the most jumping jacks initially before the time started**:
- The initial value (when time is 0 minutes) isn't given directly in either table. However, we can infer it based on their patterns. Since both have made a baseline of jumping jacks based on their starting minutes, to derive any insights accurately we consider the starting point max leading into the interpretations afterward.
- But as far as we draw conclusions **neither has a direct initial value at t=0** from provided tables. (If any movement was extrapolated, assume it's **0 for both** unless specified otherwise as base jumps).
2. **The person showing a proportional relationship between time and the number of jumping jacks**:
- To determine proportionality, we would look at the ratio of jumping jacks to time between the moments presented.
- **Kimberly's data**: It is not linear as it varies in jumps for increments.
- **Katrina's data**: Appears to show a consistent trend; 16 jumps over 2 mins equals 8 per min; 40 over 5 mins remains aligned; repeating analysis divides through confirming proportional jumps of 8/min.
- Thus, **Katrina** is more proportional in nature.
3. **The person doing the most jumping jacks per minute**:
- Calculate the jumps per minute:
- **Kimberly**:
- 3 min: \( \frac{28}{3} \) ≈ 9.33
- 8 min: \( \frac{58}{8} \) = 7.25
- 12 min: \( \frac{82}{12} \) ≈ 6.83
- 16 min: \( \frac{106}{16} \) = 6.625
- **Katrina**:
- 2 min: \( \frac{16}{2} \) = 8
- 5 min: \( \frac{40}{5} \) = 8
- 12 min: \( \frac{96}{12} \) = 8
- 20 min: \( \frac{160}{20} \) = 8
- The jumps per minute for Kimberly declined, with highest remaining at roughly 9.33, while Katrina maintained a consistent rate of 8.
- Therefore, **Kimberly** is performing at the highest single-minute rate based on the respective highest value among entries.
Based on these analyses, the responses are:
- **The person that did the most jumping jacks initially before the time started**: **Neither (or assume both start at 0 without specified initial counts)**.
- **The person showing a proportional relationship between time and # of jumping jacks**: **Katrina**.
- **The person doing the most jumping jacks per minute**: **Kimberly**.
You can navigate and place these responses in their respective areas as needed!
Match the description with the correct Function.
Function A:
A graph is shown with dot 1 being (0, -2), dot 2 is (4, -1), and dot 3 is (8, 0)
Function B:
x y
0 1
4 9
8 17
Function C: y=x+20
(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Function with the greatest rate of change
Function with the smallest y-intercept
Which function would benefit you the most if it represented your money earned per hour?
Function with the highest initial amount of money
Function A:
A graph is shown with dot 1 being (0, -2), dot 2 is (4, -1), and dot 3 is (8, 0)
Function B:
x y
0 1
4 9
8 17
Function C: y=x+20
(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
Function with the greatest rate of change
Function with the smallest y-intercept
Which function would benefit you the most if it represented your money earned per hour?
Function with the highest initial amount of money
GPT-4o mini
Let's analyze each function based on the information provided and match them with the correct descriptions.
### **Function Analysis**
1. **Function A**:
- Points: (0, -2), (4, -1), (8, 0)
- This function has a slope (rate of change) calculated as follows:
- From (0, -2) to (4, -1):
- Change in y = -1 - (-2) = 1
- Change in x = 4 - 0 = 4
- Rate of change = \( \frac{1}{4} \) or 0.25
- From (4, -1) to (8, 0):
- Change in y = 0 - (-1) = 1
- Change in x = 8 - 4 = 4
- Rate of change = \( \frac{1}{4} \) or 0.25
- The slope is constant at 0.25.
2. **Function B**:
- Points: (0, 1), (4, 9), (8, 17)
- This function has a slope:
- From (0, 1) to (4, 9):
- Change in y = 9 - 1 = 8
- Change in x = 4 - 0 = 4
- Rate of change = \( \frac{8}{4} = 2 \)
- From (4, 9) to (8, 17):
- Change in y = 17 - 9 = 8
- Change in x = 8 - 4 = 4
- Rate of change = \( \frac{8}{4} = 2 \)
- The slope is constant at 2.
3. **Function C**:
- The function \( y = x + 20 \)
- The slope is 1 (indicating a constant rate of change of 1).
### **Comparison**
- **Rate of change**:
- Function A: 0.25
- Function B: 2
- Function C: 1
- **Function B** has the greatest rate of change.
- **Smallest y-intercept**:
- Function A: y-intercept is -2
- Function B: y-intercept is 1
- Function C: y-intercept is 20
- **Function A** has the smallest y-intercept.
- **Function that benefits the most if it represented money earned per hour**:
- This would depend on the rate of change. Function B, with the highest rate of change (2), means more money is earned per hour compared to others.
- **Function B** is the most beneficial for representing money earned.
- **Function with the highest initial amount of money**:
- Function A starts at -2, Function B starts at 1, and Function C starts at 20.
- **Function C** has the highest initial amount.
### **Final Matching**:
- **Function with the greatest rate of change**: **Function B**
- **Function with the smallest y-intercept**: **Function A**
- **Which function would benefit you the most if it represented your money earned per hour?**: **Function B**
- **Function with the highest initial amount of money**: **Function C**
You can navigate and place these responses in their respective areas as needed!
### **Function Analysis**
1. **Function A**:
- Points: (0, -2), (4, -1), (8, 0)
- This function has a slope (rate of change) calculated as follows:
- From (0, -2) to (4, -1):
- Change in y = -1 - (-2) = 1
- Change in x = 4 - 0 = 4
- Rate of change = \( \frac{1}{4} \) or 0.25
- From (4, -1) to (8, 0):
- Change in y = 0 - (-1) = 1
- Change in x = 8 - 4 = 4
- Rate of change = \( \frac{1}{4} \) or 0.25
- The slope is constant at 0.25.
2. **Function B**:
- Points: (0, 1), (4, 9), (8, 17)
- This function has a slope:
- From (0, 1) to (4, 9):
- Change in y = 9 - 1 = 8
- Change in x = 4 - 0 = 4
- Rate of change = \( \frac{8}{4} = 2 \)
- From (4, 9) to (8, 17):
- Change in y = 17 - 9 = 8
- Change in x = 8 - 4 = 4
- Rate of change = \( \frac{8}{4} = 2 \)
- The slope is constant at 2.
3. **Function C**:
- The function \( y = x + 20 \)
- The slope is 1 (indicating a constant rate of change of 1).
### **Comparison**
- **Rate of change**:
- Function A: 0.25
- Function B: 2
- Function C: 1
- **Function B** has the greatest rate of change.
- **Smallest y-intercept**:
- Function A: y-intercept is -2
- Function B: y-intercept is 1
- Function C: y-intercept is 20
- **Function A** has the smallest y-intercept.
- **Function that benefits the most if it represented money earned per hour**:
- This would depend on the rate of change. Function B, with the highest rate of change (2), means more money is earned per hour compared to others.
- **Function B** is the most beneficial for representing money earned.
- **Function with the highest initial amount of money**:
- Function A starts at -2, Function B starts at 1, and Function C starts at 20.
- **Function C** has the highest initial amount.
### **Final Matching**:
- **Function with the greatest rate of change**: **Function B**
- **Function with the smallest y-intercept**: **Function A**
- **Which function would benefit you the most if it represented your money earned per hour?**: **Function B**
- **Function with the highest initial amount of money**: **Function C**
You can navigate and place these responses in their respective areas as needed!