A linear function f gives a company’s profit, in dollars, for selling x items. The company’s profit is $ 220

Question

A linear function f gives a company&rsquo;s profit, in dollars, for selling x items. The company&rsquo;s profit is $ 220 when it sells 8 items, and its profit is $ 320 when it sells 10 items. Which equation defines f ?

Accepted Answer

Choice D is correct. It&rsquo;s given that the relationship between x and f x is linear. A linear function can be written in the form f x = m x + b , where m is the slope and b is the y -coordinate of the y -intercept of the graph of y = f x in the xy -plane. Given two points on a line, x 1 , y 1 and x 2 , y 2 , the slope of the line can be found using the slope formula m = y 2 - y 1 x 2 - x 1 . It&rsquo;s given that the company&rsquo;s profit is $ 220 when it sells 8 items and the profit is $ 320 when it sells 10 items. Since f x represents the company's profit, in dollars, for selling x items, the graph of y = f x in the xy -plane passes through the points 8 , 220 and 10 , 320 . Substituting 8 , 220 and 10 , 320 for x 1 , y 1 and x 2 , y 2 , respectively, in the slope formula yields m = 320 - 220 10 - 8 , which gives m = 100 2 , or m = 50 . Substituting 50 for m , 8 for x , and 220 for f x in f x = m x + b yields 220 = 50 8 + b , or 220 = 400 + b . Subtracting 400 from each side of this equation yields - 180 = b . Substituting 50 for m and - 180 for b in f x = m x + b yields f x = 50 x - 180 . Therefore, the equation that defines f is f x = 50 x - 180 . Choice A is incorrect and may result from conceptual or calculation errors. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors.

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