Question
uestion 21 (4 points)
Listen
The graph below shows the amount of money that Zachary has saved to go towards his suit for the dance.
What is the y-intercept, and what does it represent in this situation?
What is the slope, and what does it represent in this situation?
Does this represent a proportional relationship?
If yes, identify the constant of proportionality.
If not, explain why not.
You must show ALL work to earn full credit.
the graph starts at 50 then it slope starts going up all the way to 200 as in dollars
then for the months its the same thing start at 2 for months all the way to 12
Listen
The graph below shows the amount of money that Zachary has saved to go towards his suit for the dance.
What is the y-intercept, and what does it represent in this situation?
What is the slope, and what does it represent in this situation?
Does this represent a proportional relationship?
If yes, identify the constant of proportionality.
If not, explain why not.
You must show ALL work to earn full credit.
the graph starts at 50 then it slope starts going up all the way to 200 as in dollars
then for the months its the same thing start at 2 for months all the way to 12
Answers
GPT-4o mini
To analyze the graph described, let's break down the details for y-intercept, slope, and the potential proportional relationship.
### 1. Finding the y-intercept:
The **y-intercept** is the point where the graph intersects the y-axis. Based on the information given, the graph starts at **50 dollars** when the number of months is **2**.
- **Y-intercept = 50**
- **Interpretation:** The $50 represents the initial amount of money Zachary had saved for his suit at the start of the observations (which is 2 months).
### 2. Finding the slope:
The **slope** indicates the rate of change of the dependent variable (amount saved) with respect to the independent variable (months).
From the description:
- The amount saved increases from **$50 to $200** as the months increase from **2 to 12**.
To find the slope (\(m\)), we can use the formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here:
- \(y_1 = 50\) (amount saved at \(x_1 = 2\) months)
- \(y_2 = 200\) (amount saved at \(x_2 = 12\) months)
Plugging in these values:
\[
m = \frac{200 - 50}{12 - 2} = \frac{150}{10} = 15
\]
- **Slope = 15**
- **Interpretation:** This slope means that for each month, Zachary saves an additional $15.
### 3. Does this represent a proportional relationship?
A relationship is proportional if it passes through the origin (0,0) and can be expressed in the form \(y = kx\), where \(k\) is the constant of proportionality.
- In this case, the graph does **not** start at (0,0) because it starts at (2, 50).
- Therefore, while there is a linear relationship, it is **not** a proportional relationship.
### Conclusion:
- **Y-intercept:** 50 (amount saved at the start of months; 2 months)
- **Slope:** 15 (amount saved each additional month)
- **Proportional relationship:** No
- **Reason:** The graph does not pass through the origin (0,0).
This analysis should satisfy all components of the question and demonstrate a clear understanding of the y-intercept, slope, and relationship type.
### 1. Finding the y-intercept:
The **y-intercept** is the point where the graph intersects the y-axis. Based on the information given, the graph starts at **50 dollars** when the number of months is **2**.
- **Y-intercept = 50**
- **Interpretation:** The $50 represents the initial amount of money Zachary had saved for his suit at the start of the observations (which is 2 months).
### 2. Finding the slope:
The **slope** indicates the rate of change of the dependent variable (amount saved) with respect to the independent variable (months).
From the description:
- The amount saved increases from **$50 to $200** as the months increase from **2 to 12**.
To find the slope (\(m\)), we can use the formula for slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here:
- \(y_1 = 50\) (amount saved at \(x_1 = 2\) months)
- \(y_2 = 200\) (amount saved at \(x_2 = 12\) months)
Plugging in these values:
\[
m = \frac{200 - 50}{12 - 2} = \frac{150}{10} = 15
\]
- **Slope = 15**
- **Interpretation:** This slope means that for each month, Zachary saves an additional $15.
### 3. Does this represent a proportional relationship?
A relationship is proportional if it passes through the origin (0,0) and can be expressed in the form \(y = kx\), where \(k\) is the constant of proportionality.
- In this case, the graph does **not** start at (0,0) because it starts at (2, 50).
- Therefore, while there is a linear relationship, it is **not** a proportional relationship.
### Conclusion:
- **Y-intercept:** 50 (amount saved at the start of months; 2 months)
- **Slope:** 15 (amount saved each additional month)
- **Proportional relationship:** No
- **Reason:** The graph does not pass through the origin (0,0).
This analysis should satisfy all components of the question and demonstrate a clear understanding of the y-intercept, slope, and relationship type.