Question

functions and lines
interpreting the domain and range of a linear function in context
debra is watching a movie on her laptop. as she is watching the movie, the amount of battery charge on the laptop decreases.
let y be the amount of battery charge (as a percentage), and let x be the amount of time (in minutes) since the movie began. the equation y = -0.5x + 80 models y as a function of x. as shown in the graph below, this model is accurate only from the time the movie begins until it ends. (the endpoints of the graph show when the movie begins and ends.)
graph: x-axis is amount of time (minutes) from 0 to 200, y-axis is amount of battery charge (%) from 0 to 100. a line from (0, 80) to (100, 30).
use the graph to complete each part.
(a) when the movie begins, the amount of time is minute(s). when the movie ends, the amount of time is minute(s).
each value in the select is an amount of time since the movie began.
(b) when the movie begins, the battery charge starts at %. when the movie ends, the battery charge is %.
each value in the select is an amount of battery charge.

Explanation:

Step1: Analyze part (a) - Time at start and end

The movie begins at \( x = 0 \) (since \( x \) is time since the movie began). From the graph, the end point for \( x \) (time) is at \( x = 100 \) minutes (looking at the x - axis coordinate of the end point of the line). So when the movie begins, time \( x = 0 \) minutes, and when it ends, \( x = 100 \) minutes. The set of all \( x \) values (time since movie began) is the domain of the function.

Step2: Analyze part (b) - Battery charge at start and end

The equation is \( y=- 0.5x + 80 \). When the movie begins, \( x = 0 \), so \( y=-0.5(0)+80 = 80 \%\). From the graph, the end point for \( y \) (battery charge) is at \( y = 30 \% \) (looking at the y - axis coordinate of the end point of the line). So when the movie begins, battery charge is \( 80 \% \), and when it ends, it is \( 30 \% \). The set of all \( y \) values (battery charge) is the range of the function.

Answer:

(a) When the movie begins, the amount of time is \( \boldsymbol{0} \) minute(s). When the movie ends, the amount of time is \( \boldsymbol{100} \) minute(s). Each value in the \(\boldsymbol{\text{domain}}\) is an amount of time since the movie began.
(b) When the movie begins, the battery charge starts at \( \boldsymbol{80} \% \). When the movie ends, the battery charge is \( \boldsymbol{30} \% \). Each value in the \(\boldsymbol{\text{range}}\) is an amount of battery charge.