Expert: Sherry Wallin - 3/23/2009

Question
QUESTION: Brand X wrist watches sells at a department store sell for $50 at a rate of 45 per month. The marketing department determines for every $2 decrease in price, 3 more watches sell per month. The quadratic equation that models revenue R from the watch sale is R= -6(p-25) (p+15). Graph the equation, what is the maximum revenue you can make? How many watches must you sell in order to achieve the maximum revenue?

ANSWER: Hi Jan~
    You are given the factorization of the quadratic equation that models this problem. Geometrically you will get a parabola with a quadratic equation. Thus you will get one that either opens up or down. If it opens up you will get a minimum and if it opens down you will get a maximum. Both the min and the max will occur at the vertex and the y value of the vertex will be the min or max. There are many ways to graph a parabola but probably the easiest is to put it into the form ax^+bx+c = 0 and then the  x-coordinate of the vertex is -b/2a. So let's multiply your model's equation: -6p^2+ 60p + 2250. Here a = -6,
b = 60 so -b/2a = -60/2(-6) = 5. To find y you need to substitute your x-value of the vertex back into the equation and evaluate it:
-6(5^2)+60(5) + 2250 = -150 + 300 + 2250 = 2400 so your vertex at it's maximum is (5, 2400). Notice that what you want is the number of watches you need to sell to maximize the revenue. Your y value is the maximum revenue and your x value is the number of watches.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: Thanks I had it but was not sure if that was what they were asking. How would I graph that?

Answer
There are a number of ways to graph the function but start by locating the vertex and then find the p intercepts. This is easy [they gave you the function already factored] since when R = 0 you know where the graph crosses the p axis. In other words when p - 25 = 0 and when p + 15 = 0. So now you have 3 points on the parabola: the vertex (5,2400), (25,0) and (-15, 0). You know it opens down because of the minus sign on -6p^2. Those points will give you enough and if you don't think that is enough find where the parabola crosses the R axis by setting p = 0 and solve for R. That's R = 2250 so you also have (0, 2250) for one of your points. Given that you know essentially what a parabola looks like (that it is symmetric with regards to the line x = 5, the line of symmetry) and that it opens down giving you a maximum and where it crosses both the R axis and the p axis you can do a great job graphing this function.

Math Prof