How Do You Find Special Points on a Parabola?

Let’s look at how to use formulas for a parabola to get certain important points on a parabola.

Problem For the parabola y = 2x2 + 3x – 2, locate the points below.

a. The y-intercept.

Solution At the y-intercept, the x value is zero. This means that we need to set x = 0 in the equation:

y = 2(02) + 3(0) – 2 = -2

Putting this together, the y-intercept is at (0, -2).

b. The vertex.

Solution The vertex is located using the formula   where the values of a, b, and c come from the equation. In this case, a = 2, b = 3, and c = -2. This gives an x value on the intercept of

To find the corresponding y value, put this value into the equation,

This means the vertex is at (-3/4, –25/8).

c. The x-intercepts.

Solution At the x-intercepts, the y value is zero. Putting this into the equation yields

0 = 2x2 + 3x – 2

This equation is solved with the quadratic formula,

Put the values from the equation (a = 2, b = 3, and c = -2),

The x intercepts are at (-2, 0) and (1/2, 0).

All of these points are shown in the graph of the parabola below.

What Does It Mean To Scale A Variable?

 

On many problems, the most challenging aspect is keeping track of the variables and what they represent. When the variables are scaled in hundreds, thousands, or even millions, we need to pay careful attention.

Scaling means that the quantity has been divided by some amount. For instance, we can write the amount $14,400,000 as 14.4 million dollars.

To scale $14,400,000 in millions, divide by 1,000,000 and then add the word “million” after the result. We could also scale in thousands by dividing by 1000. This gives 14,400 thousand dollars.

Let’s look at an application with scaling.

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How Do You Find Break-Even Points?

If you are looking for where the break-even points are, you must determine the quantity for which

revenue = cost

Alternately, you may find the profit by calculating

profit = revenue – cost

The problem below demonstrate these strategies starting from a demand function and a cost function. To apply either of the relationships above, you need to form the revenue function from

revenue = (price)(quantity)

where the price is given by the demand function and Q represents the quantity.

Problem The demand function for Q units of a product is given by

\displaystyle D\left( Q \right)=16-1.25Q

The cost function is given by the function

\displaystyle C\left( Q \right)=2Q+15

a. Find the revenue function R(Q).

b. Find the break-even point(s)?

c. On a graph of R(Q) and  C(Q), where do the break-even points lie?

d. Find the profit function P(Q).

e. Where do the break-even points lie on the graph of P(Q)?

Solution 1 To find the break-even point, this group of students set R(Q) = C(Q). This results in a quadratic equation. They moved all terms to one side and used the quadratic formula to find the quantities at which the revenue is equal to the cost.

Solution 2 This group of students found the profit function P(Q) first. Then they set it equal to zero to find the break-even points. Like the first solution, they also needed to use the quadratic formula.

Both techniques lead to the same break-even points and are equally valid. The only thing the second solution left out was the graph of the profit function showing the break-even points at the zeros (horizontal intercepts) of the function.

How Is Demand Related to Revenue?

I have been getting a number of questions about going from the demand function to the revenue function. This was discussed in Question 2 of Section 10.1 at the beginning of the class.

There is also a handout in the Tech 4 folder that describes the process of getting the revenue function. In Tech 4, you start by finding a line that passes through your data to get D(Q). Even though the handout in the folder uses x, I highly suggest using Q instead of x so that your mind focuses on the fact that the input represents a quantity of product. So this function takes in the quantity and outputs the corresponding price. That is what demand functions do….they relate the price of an item with the corresponding quantity that consumers demand.
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How Do You Find Quantities Related to Revenue and Cost?

In this problem, you were given revenue and cost functions like \displaystyle R(x)=-{{x}^{2}}+8x and \displaystyle C(x)=2x+5. In the parts of the problem you were asked to a) identify the correct graph, b) find the break-even quantity, c) the maximum revenue, and d) the maximum revenue. In class yesterday, the students worked in groups to solve this problem. Here is the work they put on the board.

Parts a and b

Section 10.1 Homework, Problem 7ab

Parts c and d

Section 10.1 Homework, Problem 7cd

Look pretty good but on part d, they should write the profit at x = 4 as P(4). The key in thee part is to use \displaystyle x=-\frac{b}{2a} on the quadratic revenue and profit functions to find the quantities at which the maximums are achieved. Since the parts asked you to find the maximum revenue and profit, you need to put the quantities into the functions.