Question

tracie rides the bus home from school each day. the graph represents her distance from home relative to the number of minutes since the bus left the school.

what does the slope of the graph mean?

• tracies bus travels towards her home at an average speed of \\(\frac{1}{2}\\) mile per minute.
• tracies bus travels towards her home at an average speed of 2 miles per minute.
• tracies bus travels away from her home at an average speed of \\(\frac{1}{2}\\) mile per minute.
• tracies bus travels away from her home at an average speed of 2 miles per minute.

graph: driving home with x-axis time (min) from 0 to 10, y-axis distance from home (miles) from 0 to 10. line starts at (0,9) and ends at (10,3).

Explanation:

Step1: Identify two points on the line

From the graph, we can see that when \( t = 0 \) (time in minutes), the distance \( d = 9 \) miles (point \((0,9)\)). When \( t = 10 \) minutes, the distance \( d = 4 \) miles (point \((10,4)\)).

Step2: Calculate the slope

The formula for slope \( m \) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \( m=\frac{y_2 - y_1}{x_2 - x_1} \). Here, \( x \) represents time (\( t \)) and \( y \) represents distance from home (\( d \)). So, \( x_1 = 0,y_1 = 9,x_2 = 10,y_2 = 4 \).

Substitute into the slope formula:
\( m=\frac{4 - 9}{10 - 0}=\frac{- 5}{10}=-\frac{1}{2} \)

The negative sign indicates that the distance from home is decreasing over time, which means the bus is traveling towards home. The absolute value of the slope, \( \frac{1}{2} \) mile per minute, represents the speed (rate of change of distance with respect to time) at which the bus is approaching home.

Answer:

Tracie's bus travels towards her home at an average speed of \(\frac{1}{2}\) mile per minute.