Demand function Sales records indicate that i...

Question

Demand function Sales records indicate that if Blu-ray players are priced at $250, then a large store sells an average of 12 units per day. If they are priced at $200, then the store sells an average of 15 units per day. Find and graph the linear demand function for Blu-ray sales. For what prices is the demand function defined?

Accepted Answer

Welcome back everyone. According to sales data, a bookstore can sell an average of 25 exclusive novels daily which cost $100 each. If the exclusive novel costs $125 the bookstore can sell a daily average of 20 units providers get the graph of the linear demand function. For the exclusive novel sales determine the prices at which the demand function is defined. Here, we have a graph on which we're going to sketch the graph of our linear demand function. And for our answer choices, it says this is the graph where the linear demand function is negative 1/5 of P plus 45 and the domain is between zero and 225. B says the linear demand function is 1/5 of P plus 45 and the domain is negative 225 0. C says its the demand function is negative five P plus 475 and the domain is 0 95. And the D he says the linear demand function is five P plus 475 and the domain is negative 95 0. Now let's go back to our problem here. OK. And see if we can use the information. We have to help us sketch the graph and identify the linear demand function. Now, what do we already know? Well, first, OK, let's define our variables. Let P be equal to the price of the exclusive novel. OK. And let the be equal to the average average number of exclusive novels sold per day. OK. Where we're going to use, we're going to use our price P on the X axis and we're going to put our demand or sorry, our average number of exclusive novels sold per day, D on the Y axis because the price of the exclusive novel is the independent variable. And the average number of novels sold per day is the dependent variable. What do we already know? Well, for starters, we know that the bookstore sells an average of 25 exclusive novels daily, which cost $100 each. So we already have a point P one. We can call it P one on our graph. That's equal to I say not P one. let's give this, let's just call, let's just write down the values of the point. So for this point P one, the price is $100 P one equals 100 and D one equals 25 which gives us the 0.125. So let's go to 100 here and we're going up by 10 units on the Y axis. So that's 1020 25 is gonna be somewhere right here. Ok. Next we are also told that if the exclusive novel costs 100 $25 the bookstore can sell an average of 20 units. Ok. So that gives us our, our next point where P two is 125 D two is 20. So we have the 0.1 2520. Ok. And again, notice here on our, on our 1020 30 on our X axis, we're going up by 10 units as well. So we're at 125 probably somewhere here and 20. So at this point, maybe somewhere right here. OK. I think I just lost it. Let me see if I get it back. So we know these two points will be on our graph. Now notice that we were also told that it's going to be a linear demand function. And we know that for a straight line, every straight line OK is written in the form Y equals MX plus B, the, the, the slope intercept form. So for our problem, then we can say that B as a function of P OK. That is the average number of novels as a function of the price is equal to MP plus B where M is our slope. OK. And B is our constant. So if we can figure out both of those, then we should be able to find the demand function of our graph. Now let's try to solve for M OK. Or gradient or slope. Now, the slope is going to be equal to D two minus D one, OK. The difference in Y values OK? Or be between any two points divided by P two minus P one. OK. Which we've already identified. So that's gonna be 20 minus 25 divided by 125 minus 120 minus 25 equals negative 5125 minus 100 equals 25. And the negative five divided by 25 equals negative 1/5. Therefore, our slope is negative 1/5 next to find our constant B or Y intercept. Then we can substitute any one of the given points including our slope to solve four B. OK. Now using, let's say that we were to use 100 25. OK. The soul of four be in our equation then no, for our function DFP equals MP plus B, we can replace DFP with 25. OK. M with negative 1/5. And that's being multiplied by P which is 100. OK. Plus B. Remember we're solving for B now negative fifth, multiplied by 100 equals negative 20. So if 25 equals negative 20 plus B, that means B must be equal to 25 plus 20 which is 45. So now with a slope of negative 1/5 A Y intercept of 45. Therefore, OK. That must mean that our linear demand function is equal to negative 1/5 P plus 45 know that we have our linear demand function if we go back to our problem. Uh Well, before we go back to our problem, let's also identify our domain. OK. So we have our linear demand function. What's the domain? Well, the set of all real numbers is the natural domain of a linear function. However, the price of the exclusive novel and the average number of exclusive novels sold per day should only be positive. So we know then that P must be greater than zero and D must be greater than zero. So with these values, what's going to be the value of P if D of P equals zero? OK. Now if there be OK, sorry, it's greater than zero, not equal to zero. If DFP is greater than zero, then our function OK. We're going to have negative 1/5. Oops, I may have to move a little bit over here to the right. So that negative 1/5 P plus 45 is gonna be greater than zero. That means then that negative 1/5 P, it will be greater than negative 45. And if we multiply both sides by negative five recall that when an inequality expression is multiplied by a negative number, the inequality sign should be reversed. So now that means P is going to be less than negative 45 multiplied by negative five, which is negative 225. Thus put that in red here, the domain, OK. The domain is that zero or sorry, the domain is that P is between zero and 225. So we can also say that the, the domain is 02 25. OK. So now that we have our domain, we have our linear demand function. Let's finally go and plot our graph now to plot that line, we only need two points and we've already put these two points on our graph. So all we need to do is to connect them. OK? And now when I connect them going in this direction and then let me repeat it for the other direction. OK. Then this line represents the graph of the linear demand function for the exclusive novel sales. When we look back on our answer choices, then with all the information that we have already, we can see here that a is the correct answer. Thanks for watching everyone. I hope this video helped.

Key Concepts

Linear Demand Function

Slope Calculation

Graphing the Demand Function

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