consider the graph to the right.
a. in words, describe the function shown on the graph.
b.find the slope of the graph and express it as a rate of change.
c. briefly discuss the conditions under which a linear function is a realistic model for the given situation.
c. select the correct answer below.
a. it is a good model if speed is constant for two hours and afterwards is changing.
b. it is a good model if speed is changing for two hours, then afterwards is constant.
c. it is a good model if speed is constant for 9 hours.
d. it is a good model if speed is not constant for 9 hours.
(graph: distance from home (miles) on y - axis, time (hours) on x - axis, with a line plotted from (0, 300) to (10, 0))

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Accepted Answer

# Brief Explanations:
To determine the correct answer for part c, we analyze the linear graph (distance vs. time) which implies a constant rate of change (constant speed). A linear function models motion with constant speed. Option D states it's a good model if speed is not constant for 9 hours, which is incorrect. Option A and B suggest speed changes after some time, conflicting with linearity. Option C says speed is constant for 9 hours, but the graph spans 10 hours? Wait, no—wait, the x - axis is 0 - 10 hours, and the line is linear (constant slope), meaning constant speed. Wait, no, the options: Wait, re - reading: The linear function (the graph) is a straight line, so slope (speed) is constant. So the condition for a linear model is constant speed. Let's check options:

- Option A: Speed constant for 2 hours then changing—linear model would not fit after 2 hours, so wrong.
- Option B: Speed changing for 2 hours then constant—linear model needs constant speed, so wrong.
- Option C: Wait, no, wait the graph: The line goes from (0, 300) to (10, 0)? Wait, no, looking at the graph: y - axis is distance from home (miles), x - axis time (hours). The line starts at (0, 300) maybe? Wait, no, the grid: at x = 0, y = 300? Wait, no, the first grid line on y is 200, then 400? Wait, maybe (0, 300) and (10, 0). So slope is (0 - 300)/(10 - 0)= - 30 mph (constant speed). So the linear model requires constant speed. Now, option D: "It is a good model if speed is not constant for 9 hours"—no, that's wrong. Wait, maybe I misread. Wait the options:

Wait the original options:

A. It is a good model if speed is constant for two hours and afterwards is changing.

B. It is a good model if speed is changing for two hours, then afterwards is constant.

C. It is a good model if speed is constant for 9 hours.

D. It is a good model if speed is not constant for 9 hours.

Wait, no, maybe the graph is from (0, 300) to (9, 0)? Wait, the x - axis is 0 - 10, but the line ends at x = 9? Wait, the rightmost point is at x = 9? Maybe. So the time period is 9 hours? Wait, the x - axis labels: 0, 2, 4, 6, 8, 10. The line goes from (0, 300) to (9, 0)? So the time duration is 9 hours. A linear function (constant slope) implies constant speed. So the condition for the linear model is that speed is constant (since slope is constant). Wait, no—wait the question is "the conditions under which a linear function is a realistic model". A linear function for distance - time means constant speed (since distance = speed×time + initial distance, linear in time). So if speed is constant, the linear model works. Now, let's re - evaluate options:

Wait, maybe the correct option is D? No, that says speed is not constant for 9 hours. Wait, maybe I made a mistake. Wait, the key is: a linear function has a constant rate of change (slope). So for the distance - time graph, linear means constant speed. So the linear model is good when speed is constant. Now, looking at the options:

Option D: "It is a good model if speed is not constant for 9 hours"—no, that's contradictory. Wait, maybe the options are misprinted? Wait, no, let's think again. Wait the graph: the line is straight, so slope is constant (constant speed). So the linear model is appropriate when speed is constant. Now, the options:

A: speed constant for 2 hours then changing—linear model would not fit after 2 hours, so wrong.

B: speed changing for 2 hours then constant—linear model needs constant speed from the start, so wrong.

C: speed constant for 9 hours—if speed is constant for 9 hours, then the linear model (which has constant slope) would fit for those 9 hours. Wait, but the graph spans up to 10 hours? Wait, maybe the x - axis is up to 9 hours? The rightmost point is at x = 9? So the time period is 9 hours. So if speed is constant for 9 hours, the linear model (straight line) is good. Option D says speed is not constant for 9 hours, which would mean the slope (speed) changes, so linear model is bad. So the correct answer is C? Wait, no, wait the option C: "It is a good model if speed is constant for 9 hours." Because a linear function has constant slope (constant speed), so if speed is constant for 9 hours, the linear model (the straight line) is a good fit for those 9 hours. Option D says speed is not constant for 9 hours, which would mean the slope changes, so linear model is not good. So the correct answer is C? Wait, no, I'm confused. Wait, let's re - read the question: "Briefly discuss the conditions under which a linear function is a realistic model for the given situation." The given situation is the graph (distance vs. time, linear). So the linear graph implies constant speed. So the condition is that speed is constant. Now, looking at the options:

Option C: "It is a good model if speed is constant for 9 hours." If speed is constant for 9 hours, then distance changes linearly with time (since distance = speed×time + initial distance), so the linear model is good. Option D: "It is a good model if speed is not constant for 9 hours." If speed is not constant, distance - time graph is not linear, so D is wrong. Option A: speed constant for 2 hours then changing—linear model would only fit the first 2 hours, not the whole graph, so wrong. Option B: speed changing for 2 hours then constant—linear model would not fit the first 2 hours, so wrong. So the correct answer is C? Wait, but the x - axis is up to 10 hours. Wait, maybe the line ends at x = 9, so the time period is 9 hours. So if speed is constant for 9 hours, the linear model (the straight line) is a good fit. So the correct option is C.

# Answer:
C. It is a good model if speed is constant for 9 hours.

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