Determine x-intercept and y-intercept

Question

A phone company charges for services according to the formula: C = 27 + 0.09n, where n is the number of minutes talked, and C is the monthly charge, in dollars.

The rate of change in this equation is:

The initial value in this equation is:

Answer 1

y = mx + b, where m = the slope (rate of change) and b = the y-intercept (the initial value).

C = 27 + 0.09n

C = 0.09n + 27

If I talk no minutes...

C = 0.09(0) + 27

C = 27

Hopefully you can see without doing any calculations that the constant (b) is the initial value.

The coefficient of the term with the variable is the rate of change, the cost per minute in this case.

Soooo, again, I am charged $27 even if I talk no minutes. I am also charged 9 cents for every minute that I do talk. If I talk for 2 hours, I am charged 27 + 0.09(120) = $37.80. If I talk for just 5 minutes, I'm charged 27 + 0.09(5) = $27.45.

Since the title is "find the x-intercept and y-intercept" I point out that along the y-axis, x = 0, so, again, the y-intercept is $27.

Along the x-axis, y = 0. There is no way to be charged nothing, so there is no x-intercept.

Answer 2

The initial condition is when n=0. So the initial charge per month is C(0) = 27 dollars (the y-intercept).

The rate of change for the charges over time is the slope which is 0.09 dollars/minute (dollars per minute). This can be found without calculus by noting that units for 27 is dollars, thus 0.09n has to result in dollars as well. Sine n is the number of minutes, then the units for 0.09 has to be dollars/minute (a rate of change in charges) in order for the minutes to cancel out each other and to be left dollars t be added to 27.

Another way to find the the rate of change for charges, is to use calculus to take the time derivative (w.r.t. the variable time variable n) of the equation C = 27 + 0.09n, which results in just 0.09n.