Question 126·Hard·Linear Functions
An indie record label models the relationship between the number of times a new single is downloaded, , and the net revenue, , in dollars, with the equation
According to this model, what is the best interpretation of ?
For questions about interpreting parts of a linear model, first identify what each symbol and coefficient represents (variables, units, and whether it is a rate or a starting value). Recognize that the coefficient of the variable is the rate of change (slope), while the constant term is the value when the variable is zero (the starting amount). Plug in the key value (often 0 for the intercept) to see what the expression equals, then match that result to the answer choice that correctly describes it in words using the context and units given.
Hints
Focus on the constant term
Look at the equation . Which part of the equation does not change when the number of downloads changes?
Try plugging in a simple value
What is when ? Calculate that value and think about what it means in this situation.
Interpret the sign of the value
If the net revenue comes out negative at 0 downloads, what does that say about the label’s financial position before any downloads occur?
Match to the answer choice
Once you know what happens at 0 downloads, choose the option that describes that situation in words.
Desmos Guide
Graph the revenue model
In Desmos, type R(d) = 1.29d - 350 or f(d) = 1.29d - 350 to define the revenue as a function of downloads. You will see a straight line with a negative vertical intercept.
Evaluate revenue at zero downloads
In the expression list, add a new line and type R(0) or f(0). Look at the output value: this is the net revenue when there are 0 downloads, and it tells you the starting financial position before any sales.
Connect the graph to the context
On the graph, notice where the line crosses the vertical axis (when ). Think about what that y-value means for the label’s net revenue at the very beginning and choose the option that describes that situation.
Step-by-step Explanation
Recognize the structure of the equation
The equation is linear:
This matches the slope–intercept form , where:
- (here ) is the slope, or rate of change per download.
- (here ) is the vertical intercept, the value of when .
Find what happens at 0 downloads
To interpret the constant term , set :
.
So, when there are 0 downloads, the model says the net revenue is −350 dollars.
Interpret a negative revenue in context
Net revenue is income minus costs. A value of −350 dollars means the label is 350 dollars below zero in net revenue at the point of 0 downloads, which corresponds to money already spent (a fixed upfront cost) before any sales happen. This matches answer choice D) The label spent 350 dollars before any downloads were made.