Question

the speed of a car traveling on a highway is being recorded once per second for two minutes. during this time interval, the car gradually speeds up slightly to pass another vehicle, then the car returns to its original speed. the recorded speed of the car with respect to time can be modeled by linear, quadratic, and exponential functions. for each of the three models, their residuals are small and are without pattern. which of the following conclusions is best? a a linear model is best based on contextual clues. b a quadratic model is best based on contextual clues. c an exponential model is best based on contextual clues. d contextual clues fail to help in selecting a model for this contextual situation.

Explanation:

• A linear function has a constant rate of change (either increasing or decreasing at a steady pace). But in this case, the car first speeds up and then returns to original speed. A linear function cannot model the "speeds up - returns" behavior as it doesn't have a change in the direction of the rate of change (from increasing to decreasing).
• An exponential function has a rate of change that is proportional to the function's current value. It either shows rapid growth (if base > 1) or decay (if \(0<\text{base}<1\)) and doesn't typically model a "speed - up then return to original" scenario well in a simple, common - sense way for a car's speed over a short time interval like 2 minutes.
• A quadratic function \(y = ax^{2}+bx + c\) (\(a

eq0\)) has a parabolic shape. If \(a<0\), it has a maximum point. This can model the situation where the car's speed first increases (as it accelerates to pass) and then decreases (as it returns to the original speed), similar to the shape of a parabola opening downwards.

Answer:

B. A quadratic model is best based on contextual clues.