Question

The graph below shows a company's profit f(x), in dollars, depending on the price of erasers x, in dollars, being sold by the company:

graph of quadratic function f of x having x intercepts at ordered pairs 0, 0 and 8, 0. The vertex is at 4, 270

Part A: What do the x-intercepts and maximum value of the graph represent? What are the intervals where the function is increasing and decreasing, and what do they represent about the sale and profit?

Part B: What is an approximate average rate of change of the graph from x = 1 to x = 4, and what does this rate represent?

Answer 1

Answer:

A. x-intercepts: prices at with profit is zero

    maximum: price at which profit is a maximum

    increasing: (-∞, 3); decreasing: (3, ∞) — profit increases with increasing pen price below $3, decreases with increasing pen price above $3

 B. -26 2/3; loss of profit for each dollar increase in pen price

Step-by-step explanation:

Part A:

The definition of the function tells you what points on the graph represent. The x-intercept at x=0 tells you the company's profit will be zero when the price of pens sold by the company is zero.

The x-intercept at x=6 tells you the company's profit will be zero when the price of pens sold is $6.

The maximum value of 120 at x=3 tells you the company's profit will be $120 when the price of pens sold is $3.

The graph goes "uphill" for x < 3, so the function is increasing for x < 3. The function goes "downhill", so is decreasing, for x > 3. This tells you profit increases with pen price when the price is below $3, and decreases with pen price above $3.

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Part B:

The equation of the function describing the graph is ...

 f(x) = -120/9x(x -6)

so, at x=5, the value is ...

 f(5) = -120/9(5)(-1) = 600/9 = 66 2/3

Then the average rate of change from x=3 to x=5 is ...

 m = (f(5) -f(3))/(5 -3) = (66 2/3 -120)/2 = (-53 1/3)/2 = -26 2/3

This represents the average change in profit for each dollar increase in the price at which pens are sold.