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.