Question

After a month of driving from neighborhood to neighborhood and walking door-to-door, she figures out that her weekly earnings are approximately a linear function of the number of doors she knocks on.

She writes the equation of the function like this: E(x) = 7x - 25, where x is the number of doors she knocks on during the week and E(x) is her earnings for the week in dollars.

Which of the following are reasonable interpretations of the y-intercept o

Answer 1

A. She can earn $7 per week even if she does not knock on any doors.
B. Her expenses are $25 per week.
C. If she does not knock on any doors at all during the week, she will lose $25.
D. She will lose $7 per week if she does not knock on any doors.

Answer 2

Looking at the function E(x) = 7x - 25 we can get a pretty good idea of the graph (first realise that the y-values obtained for any value x, is the amount of money she makes or loses):

1) The graph is steep as 7 is relatively large gradient wise: You need a big change in y-values produced by a small change in x values. This means she can quickly start covering her losses as the break-even point (the point where she stops making a loss and covering costs) is equal to => 7x - 25 = 0 which is equal to x = 25/7 which means when she has knocked on four doors her costs are covered (and she has actually already made a bit of profit).

2) The y-intercept is -25 which means she starts out at a loss of -25.

Now we know A and D are not viable options because E(0) = 7(0) - 25 = - 25 which means either she has expenses of 25 or she will make a loss of 25. Differentiating between the two can be a bit tricky without some accounting knowledge: your expenses are never fixed like that unless you are dealing with manufacturing which is a whole 'nother ball game.

But, looking closely at the question and considering what the equation is for you can realise it is C: "E(x) is her earnings for the week in dollars". This means that when
E(x) = -25 then she has a loss of 25 dollars. Just focusing on the fact that the y-axis gives the amount of money she makes you can realise that when it is negative she is losing money. Remember, however, to ignore values of y when x is less than zero in this function as she can't knock of -1 doors.