Question

in an earlier lesson, we saw this graph that shows the percentage of all garbage \boxed{} between 1991 and 2013 in the united states that was recycled.
1 sketch a linear function that models the change in the percentage of garbage that was recycled between 1991 and 1995. for which years is the model good at predicting the percentage of garbage that is recycled? for which years is it not as good?
2 pick another time period to model with a sketch of a linear function. for which years is the model good at making predictions? for which years is it not very good?

Explanation:

Step1: Analyze 1991 - 1995 Data

First, identify the data points between 1991 and 1995. The points around this period (from the graph: ~1990 - 1995) show a trend. To sketch a linear model, we can take two points: say, (1991, ~16%) and (1995, ~21%). The linear function will have a slope calculated as $\frac{21 - 16}{1995 - 1991}=\frac{5}{4} = 1.25$ percentage points per year. The equation would be $y - 16 = 1.25(x - 1991)$, where $y$ is the percentage recycled and $x$ is the year.

Step2: Evaluate Model Goodness

• Good Prediction Years: The model is good for years close to 1991 - 1995, like 1992 - 1994, as the data points are relatively close to the linear trend here.
• Not Good Prediction Years: Years far from 1991 - 1995, like before 1990 (e.g., 1989) or after 1995 (e.g., 2000, 2010), because the data deviates more from the linear trend of 1991 - 1995 as we move away from this period.

Step3: Another Time Period (e.g., 2005 - 2010)

• Sketch Linear Model: Take points (2005, ~23%) and (2010, ~25%). Slope is $\frac{25 - 23}{2010 - 2005}=\frac{2}{5}=0.4$ percentage points per year. Equation: $y - 23 = 0.4(x - 2005)$.
• Good Prediction Years: 2006 - 2009, as data points in 2005 - 2010 are close to linear.
• Not Good Prediction Years: Years far from 2005 - 2010, like 1990 or 2015, as the trend changes (e.g., after 2010, the percentage starts to decrease, which the 2005 - 2010 linear model (increasing) won't capture well).

Answer:

For 1991 - 1995 Linear Model:

• Sketch: A line connecting points around 1991 - 1995 (e.g., from ~16% in 1991 to ~21% in 1995).
• Good Prediction Years: 1992 - 1994 (years near 1991 - 1995).
• Not Good Prediction Years: Before 1990 (e.g., 1989) and after 1995 (e.g., 2000, 2010).

For Another Period (e.g., 2005 - 2010):

• Sketch: A line from ~23% (2005) to ~25% (2010).
• Good Prediction Years: 2006 - 2009 (years near 2005 - 2010).
• Not Good Prediction Years: Before 2000 (e.g., 1995) and after 2010 (e.g., 2015, as the trend reverses).

(Note: The specific years can vary slightly based on graph interpretation, but the key is the proximity to the modeled period for good predictions and deviation for poor ones.)