Quadratic Functions

Question

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

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 = 2 to x = 5, and what does this rate represent? (3 points)

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

Answer 1

I've recreated the graph based on the information in your problem:

A) The x-intercepts are the places on the x-axis where the value of the graph is zero. So, in this case, when pencils are sold for $0.00 there is $0.00 profit for the company and when pencils are sold for $10.00 there is $0.00 profit for the company. This most likely means that when they price the pencils at $10, nobody will buy any of them. But, when they price pencils at $5.00, they get their maximum profit of $160. The function is increasing between zero and 5 ($0.00 and $5.00) and decreasing between 5 and 10 ($5.00 and $10.00). This means that when they increase the price of pencils above $0.00, their profits will increase until the price reaches $5.00. Then when they increase their price above $5.00, their profits will decrease until the price reaches $10.00 where the profit will be zero.

B) To find the function, write it in factored form: y = a(x)(x - 10) and then plug in x = 5 and y = 160 to solve for "a":

160 = a(5)(5 - 10)

160 = -25a

a = -6.4

So the function is y = -6.4(x)(x - 10) where "y" is the company profits and "x" is the pencil price.

y = -6.4(x)(x - 10)

y = -6.4(x² - 10x)

y = -6.4x² + 64x

when x = 2, the profit is: -6.4(2)² + 64(2) = $102.40

when x = 5, the profit is: -6.4(5)² + 64(5) = $160.00

The average rate of change is (y2 - y1)/(x2 - x1) = (160.00 - 102.40)/(5 - 2) = 57.6/3 = 19.2 which means that, on average, for every dollar the pencil price increases, the profit increases by $19.20

C) The domain is the possible values of the price of the pencils. So it would be between $0.00 and $10.00. It means they could realistically price their pencils anywhere between and including $0.00 and $10.00 (although, they would be wisest to price them at $5.00).

Answer 2

A quadratic equation in vertex form is:

y = a(x - h)² + k

Where:

Vertex is: (h, k)

Substituting 5 for h, 160 for k

y = a(x - 5)² + 160

And (0,0) satisfies the function, so substituting for x and y

yields a = -32/5 and the function is:

y = -(32/5)(x - 5)² + 160

The graph is shown below:

Part A:

The x-intercepts represent the zeros of the function (the values where the pencil price generates exactly zero profit). The maximum value (at the vertex) shows the optimal pencil price ($5) generating the maximum profit ($160).

The function (and profit) is increasing as pencil prices rise to $5, and then the function decreases as pencil prices rise above $5.

Part B:

Rate of change form x= = 2 to x = 5

Rate of change = (f(5) – f(2))/(5 - 2)

Rate of change = (160 – ((-32/5)(2 – 5)² + 160)/(5 – 2)

Rate of change = (160 – 102.4 )/3

Rate of change = 57.6/3 = 19.2

Part C:

The domain (pencil price) is limited by the fact a profit must be greater than or equal to zero.   Therefore, the pencil price, x, must be:

0 <= x <= 10