Question

PLEASE HELP
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?

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?

Part C: Describe the constraints of the domain.

Answer 1

Answer:

See below ↓

Step-by-step explanation:

Part A

• The x-intercepts represents when the profit of the company is 0 dollars. It means they are making no profit when selling the pens at that price

Part B

• y-value when x = 3 ⇒ 120
• y -value when x = 5 ⇒ 60
• rate of change = Δy/Δx = -60/2 = -30
• This rate represents the depreciating profit of the company when increasing the price of a pen from $3 to $5

Part C

• Domain of the graph is only true for positive values of y
• Domain ⇒ 0 ≤ x ≤ 6

Answer 2

Answer:

Part A

The x-intercepts are when f(x) = 0.

Since x represents the price of pens, and f(x) represents the company's profit, the x-intercepts give the price of the pens when the profit is zero.

⇒ the profit is zero when the price of pens are $0 and $6

The maximum value of the graph (vertex) represents the cost of pens when the profit is at its highest.  Therefore, the optimal amount to price the pens to get the maximum profit.

⇒ maximum profit of $120 occurs when the pens are priced at $3

Part B

$\sf average \ rate \ of \ change=\dfrac{change \ in \ y}{change \ in \ x}$

From inspection of the graph, f(3) = 120 and f(5) ≈ 60

$\sf \implies average \ rate \ of \ change=\dfrac{60-120}{5-3}=-30$

This means that as the price of the pens increases by $1 (between $3 and $5), the average profit decreases by approximately $30.

Part C

The domain is the set of input values (x-values).  As the price of a pen cannot be less than zero, the domain will be x ≥ 0

Unless the company is happy to make a loss, the restricted domain should be 0 ≤ x ≤ 6