Question

graphs, functions, and sequences
drawing a graph to fit a narrative
greg is delivering a pizza to mark’s house and drives at a constant speed toward the house. then, he hits a traffic jam and stops for several minute
he starts up again and drives toward the house.
using three segments, sketch a possible graph that shows greg’s distance to mark’s house versus time.
distance to marks house
time

Explanation:

Step1: Analyze the first segment

Greg drives at a constant speed toward the house. So, distance to the house decreases linearly with time. The graph here is a straight line with a negative slope (since distance is decreasing as time increases). Let's assume the initial distance is \( d_0 \) (some positive value) and time starts at \( t = 0 \). So, from \( t = 0 \) to \( t = t_1 \) (when he hits traffic), the line goes from \( (0, d_0) \) to \( (t_1, d_1) \) where \( d_1 < d_0 \) and the slope is \( \frac{d_1 - d_0}{t_1 - 0} \) (negative, constant speed).

Step2: Analyze the second segment

He hits a traffic jam and stops for several minutes. When stopped, time increases but distance to the house remains constant. So, the graph here is a horizontal line (slope = 0) from \( t = t_1 \) to \( t = t_2 \) (when traffic clears), with \( d(t) = d_1 \) for \( t_1 \leq t \leq t_2 \).

Step3: Analyze the third segment

He starts up again and drives toward the house. Again, distance decreases linearly with time (constant speed, same or different? The problem says "drives toward the house" – we can assume constant speed, so another straight line with negative slope (maybe same or different, but still negative) from \( t = t_2 \) to \( t = t_3 \) (when he reaches the house, distance = 0), going from \( (t_2, d_1) \) to \( (t_3, 0) \) with slope \( \frac{0 - d_1}{t_3 - t_2} \) (negative, constant speed).

To sketch:

1. Draw the first line: Start at the top of the y - axis (distance) at time 0, go down to some point (time \( t_1 \), distance \( d_1 \)).
2. Draw the second line: Horizontal from \( (t_1, d_1) \) for a few units (time increase, distance same).
3. Draw the third line: From \( (t_2, d_1) \) down to the x - axis (distance 0) at time \( t_3 \).

Answer:

The graph has three segments: (1) A line with negative slope (distance decreasing over time) from the initial time - distance point, (2) A horizontal line (distance constant, time increasing) during the traffic jam, (3) A line with negative slope (distance decreasing over time) until distance reaches 0. (Sketch as per the above analysis: first decreasing line, then horizontal, then decreasing line to zero.)