Question

example 4: the graph of the function ( y = f(x) ) in the ( xy )-plane always has a negative rate of change. which of the following definitions for the variables ( x ) and ( y ) would best model the function ( f )?
(a) ( x = ) the age, in years, of a young child; ( y = ) the height, in inches, of the young child
(b) ( x = ) the total number of points scored in a basketball game; ( y = ) the time remaining, in seconds, in the game.
(c) ( x = ) the time, in seconds, since a ball was thrown straight up in the air; ( y = ) the height, in feet, of the ball
(d) ( x = ) the radius, in meters, of a circle; ( y = ) the area, in square inches, of the circle

Explanation:

Step1: Analyze Option A

A young child's height generally increases with age, so the rate of change of height with respect to age is positive. So A is incorrect.

Step2: Analyze Option B

As the total number of points scored in a basketball game (\(x\)) increases, the time remaining in the game (\(y\)) decreases. So the rate of change of \(y\) with respect to \(x\) is negative, which fits the function having a negative rate of change.

Step3: Analyze Option C

When a ball is thrown straight up, its height first increases (as it moves up) and then decreases (as it falls down). So the rate of change of height with respect to time is not always negative. So C is incorrect.

Step4: Analyze Option D

The area of a circle \(A=\pi r^{2}\), as the radius (\(x\)) increases, the area (\(y\)) increases. So the rate of change of area with respect to radius is positive. So D is incorrect.

Answer:

B. \( x =\) the total number of points scored in a basketball game; \( y =\) the time remaining, in seconds, in the game.