Digital music and CD Algebraic task

Question

After analyzing the data, the sales manager of the company modeled the difference in sales using the cubic function s(x)=x^3-13x^2+44x-32 where x represents the number of years since 2005. Construct a sketch of the graph of s(x) over the domain 0< x < 10. In which year(s) was the total amount in sales of digital music equal to the total amount in sales of CDs? Provide evidence to support your answer.

Part B The sales manager also wanted to model the difference in sales, in hundreds of dollars, between the total annual country music sales and the total annual hip-hop music sales. She found that the total annual sales of these two types of music were equal in the same years that the total sales of digital music equaled the total sales of CDs found in Part A. The manager stated that the model for this data must also be a cubic function. Is her conjecture correct? Provide evidence and a graph to support your answer.

Answer 1

Hey Morgan,

In calculus, you will eventually learn more precise ways of drawing graphs like this, but I don't want to overcomplicate this for Algebra 2. Here's what I would do:

A) Just plug in x = 0 through x = 10 to find s(x) at each point. To sketch this, use your max and min values to create an appropriate scale so all your points will fit. Then, just plot each point and connect the dots with curved lines. The problem says that s(x) is the difference in sales. So that means that the sales are equal whenever s(x) = 0.

B) What a strange question. Assuming the shape of a graph based on a couple data points is never appropriate, especially when there is no physical law at play. There are many kinds of graphs that can pass through a small number of points. As a way to prove this, I suppose you could sketch some non-cubic functions that cross the x-axis 3 times between 0 and 10. Many trig functions can do this, as can 4th-order equations, 5th-order equations, etc.

Hope this helps,

Chris