Question

a graph is drawn to represent a relationship between distance, (d), and time, (t).

option 1: each point shows the distance, in feet, remaining to the finish line of a race (t) seconds after a runner begins the final stretch and runs at a constant speed.

option 2: each point shows the distance, in feet, covered (t) seconds after a brake is applied, where the speed of the car decreases by a constant factor.

option 3: each point shows the distance, in feet, traveled by an airplane on a runway by gradually increasing speed (t) seconds after being approved for takeoff.

option 4: each point shows the distance, in feet, above the ground of a falling object after (t) seconds, where the falling speed increases as time increases.

Explanation:

Analyze the graph's characteristics

Using the Linear Modeling and Constant Rate of Change knowledge points

• The graph shows discrete points representing a relationship between distance \(d\) (vertical axis) and time \(t\) (horizontal axis).
• The points lie perfectly along a straight line with a negative slope.
• This indicates that distance \(d\) decreases at a constant rate as time \(t\) increases.

Evaluate the first option

Using the Linear Modeling and Constant Rate of Change knowledge points

• The first option describes a runner's distance to the finish line of a race \(t\) seconds after starting the final stretch at a constant speed.
• As time \(t\) increases, the remaining distance to the finish line decreases at a constant rate (constant speed).
• This perfectly matches a linear relationship with a negative slope.

Evaluate the second option

Using the Linear Modeling and Constant Rate of Change knowledge points

• The second option describes a car's speed decreasing by a constant factor.
• A constant factor change represents exponential decay, which would produce a curved graph rather than a straight line.

Evaluate the third option

Using the Linear Modeling and Constant Rate of Change knowledge points

• The third option describes an airplane on a runway gradually increasing its speed.
• An increasing speed means the distance traveled per unit of time is increasing, which would produce a curve with an increasing slope (quadratic/exponential growth).

Evaluate the fourth option

Using the Linear Modeling and Constant Rate of Change knowledge points

• The fourth option describes a falling object whose falling speed increases as time increases.
• An increasing falling speed means the distance above the ground decreases at an accelerating rate, which would produce a downward-curving quadratic path.

Answer:

• (A) Each point shows the distance, in feet, remaining to the finish line of a race t seconds after a runner begins the final stretch and runs at a constant speed. (Correct answer)
• (B) Each point shows the distance, in feet, covered t seconds after a brake is applied, where the speed of the car decreases by a constant factor.
• (C) Each point shows the distance, in feet, traveled by an airplane on a runway by gradually increasing speed t seconds after being approved for takeoff.
• (D) Each point shows the distance, in feet, above the ground of a falling object after t seconds, where the falling speed increases as time increases.