Question

. A new cell phone comes on the market. Sales (S, in millions) increase at a steady rate for several months then decrease at about the same rate. This can be modeled by the function

S(m)= -0.375|m-12|+15

(a) Graph the function, using correct labels and units.
(b) What is the vertex? What does the vertex mean in terms of the problem?
(c) What is the rate of change of the sales?

Answer 1

(a) see below for a graph

(b) the vertex is (months, sales) = (12, 15); sales is $15M at 12 months

(c) .375 million per month increasing and decreasing

Explanation:

(a) You can read the vertex from the equation of the function. The equation is of an absolute value function translated so its vertex is at (12, 15), and vertically scaled by a factor of -0.375. The negative scale factor means the graph will open downward.

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(b) Since graph is of sales in millions versus months, the meaning of the vertex at (12, 15) is that sales is $15 millions 12 months after the phone comes on the market.

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(c) If the function were written as a piecewise function, the coefficient of x would be +0.375 for x < 12 and -0.375 for x > 12. The "rate of change" is 0.375 millions per month both going up and coming down.

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Translation of a point on the graph of f(x) by "h" horizontal units and "k" vertical units changes the function to f(x -h) +k. That is, if you can identify the function f(x), you can read the translation from the expression f(x -h) +k. For the absolute value function |x|, the vertex is normally (0, 0). Translating it to (h, k) makes the expression be |x -h|+k, the form you see in this problem. It's not a mystery. It's just pattern matching.