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

Graph of quadratic function f of x having x intercepts at ordered pairs 0, 0 and 6, 0. The vertex is at 3, 120.

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? (4 points)

Part B: What is an approximate average rate of change of the graph from x = 3 to x = 5, and what does this rate represent? (3 points)

Part C: Describe the constraints of the domain. (3 points)

A) x intercepts are the breakeven points, zero profit

Increasing: 0《x < 3

Profit increases as selling price increases

Decreasing: 3 < x《6

Profit decreases as selling price increases

B) -80/3

Per $1 increase in Selling price decreases profit by $80/3

C) 0《x《6

Step-by-step explanation:

y = a(x - h)² + k

y = a(x - 3)² + 120

0 = a(0 - 3)² + 120

9a = -120

a = -40/3

y = -(40/3)(x - 3)² + 120

A) x intercepts are the breakeven points, zero profit

Cost = sales

Maximum value of the graph is the maximum profit the firm can make, which is $120

Uptil the vertex, function increases. After that, it decreases

Increasing: 0《x < 3

Profit increases as selling price increases

Decreasing: 3 < x《6

Profit decreases as selling price increases

B)

x = 3,

f(x) = 120

x = 5,

f(x) = -(40/3)(5 - 3)² + 120 = 200/3

Average rate:

(200/3 - 120) ÷ (5 - 3)

-80/3

C) x》0 because sales can't be negative

x《6 because there's a loss if the selling price is more than $6

Therefore,

0《x《6

see below

Step-by-step explanation:

Part A  The x intercepts represent when the profit is zero.  The maximum value is at the vertex and is where the maximum profit occurs.  The function is increasing until it reaches the vertex and is decreasing after  the vertex.  The profit is increasing until it reaches its peak at the vertex and is decreasing after the vertex until they reach the zero.  Profits are negative to the left of the first zero and to the right of the second zero

(-inf to 0) negative profit   but increasing  (0 to 3) increasing profit  3 maximum profit   (3 to 6) decreasing profit but still positive  (6 to negative infinity) decreasing negative profit

Part B  The approximate rate of change is  how much the profit decreases when we go from $3 of the item to $5.  The graph is required to calculate this

f(5) - f(3)

5-3

Approximate the values of

f(5) and f(3) from your graph

Part C  the domain is constrained by x at =0, we cannot sell pay people to take the item.  Business wise, we are constrained at x=6 because we do not want to make a negative profit

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step-by-step explanation: