The graph below shows a company's profit f(x) in dollars

Question

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

 

The graph is a parabola shaped like a frown. The x-intercepts are 0 and (6,0). The vertex is 120. 

 

Part B: What is an appropriate average rate of change of the graph from x=3 to x= 5 and what does the rate represent? 

Answer 1

I am not entirely sure what your question is, but I will walk you through the concept and the solution.

If you think about a going on a road trip in a car, the average rate of change is the distance you travel divided by how long it took you to cover that distance. The distance traveled is actually displacement because you just compare where you end up to where you start. Graphically, this means that if you are finding the average rate of change from x=3 to x=5, you need to figure out the change in y-values between x=3 and x=5 and then divide that by the horizontal change, which is 5-3 or 2.

If the parabola has x-intercepts at x=0 and x=6, we know the vertex is halfway between. So, the parabola has vertex (3, 120). This means that the equation for the parabola is f(x)=a(x-3)²+120 for some unknown value a. We can find a by plugging one of the x-intercepts, let's use (0,0), into the equation and solving for a.

So,

0=a(0-3)²+120

-120 = 9a

a = -120/9 = -40/3

Thus, the parabola has equation f(x)=(-40/3)(x-3)²+120

We already knew that at x=3, f(3)=120

Using the equation, f(5)=(-40/3)(5-3)²+120=(-40/3)(4)+120 = -160/3 + 120 = 200/3

Thus, calculating the average rate of change between x=3 and x=5, we get

(f(5)-f(3))/(5-3) = (200/3 - 120)/2 = (-160/3)/2 = -80/3

So, the average rate of change is -80/3.