Eva has purchased 200 songs from the internet. She plans to download an equal number of songs on her music player each week for 5 weeks. The graph shows the number of songs left to download, y, for a certain number of weeks, x:
A graph titled Song Downloading shows Number of Weeks on x-axis and Number of Songs Left to Download on y-axis. The x-axis scale is shown from 0 to 5 at increments of 1, and the y-axis scale is shown from 0 to 280 at increments of 40. A straight line joins the ordered pairs 0, 200 and 1, 160 and 2, 120 and 3, 80 and 4, 40 and 5, 0.
Part A: What is the rate of change and initial value of the function represented by the graph, and what do they represent in this scenario? Show your work to find the rate of change and initial value. (6 points)
Part B: Write an equation in slope-intercept form to model the relationship between x and y. (4 points)
Part A:
To find the rate of change, we need to calculate the change in y divided by the change in x. From the graph, we see that the initial value is 200 songs and after 1 week, there are 160 songs left to download.
Change in y = 200 - 160 = 40
Change in x = 1 - 0 = 1
Rate of change = Change in y / Change in x = 40 / 1 = 40
The rate of change of 40 represents the number of songs Eva downloads each week.
The initial value is 200, which represents the total number of songs she purchased.
Part B:
We can write the equation in slope-intercept form as:
y = mx + b
where m is the slope (rate of change) and b is the y-intercept (initial value).
From Part A, we know that the slope (m) is 40 and the initial value (y-intercept, b) is 200.
Therefore, the equation in slope-intercept form is:
y = 40x + 200