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The manager of a weekend flea market knows from past experience that if he charges $ x $ dollars for a rental space at the market, then the number $ y $ of spaces he can rent is given by the equation

$ y = 200 -4x $.

(a) Sketch a graph of the linear function. (Remember that the rental charge per space and the number of spaces rented can't be negative quantities.)

(b) What do the slope, the y-intercept, and the x-intercept of the graph represent.

a. see the graph of the equation in the full solution.

b. See the grahp and the explanation above

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Johns Hopkins University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

here we have the equation, which relates the charge per space rented with the number of spaces rented, and we want to start by sketching the graph. So I'm going to find a couple key points on the graph and plot those and connect them to make a line. If X is zero, that would mean they don't charge anything per space. Why will be 200? They would rent 200 spaces, 200 spaces for free, and if why is zero? Then X would end up being 50. So that means that if they charge $50 per space, they won't rent any of them. And these air the intercepts. So let's go ahead and plot these points and will get a feel for what the line lives like. Now I'm not going to extend my line into any other quadrants. It's going to stay in quadrant one because you can't have a negative number of spaces rented, and you wouldn't collect a negative amount of rent. You wouldn't charge a negative amount per space. All right, so there's our graph now. What we want to do is think about the slope and the Y intercept and the X intercept. We talked about the intercepts a little bit already. So what's the slope when you look at the equation that was given, you see that the slope is negative. For what are the units on slope? So remember that slope is rise over run. So we would have the units on. Why? Divided by the units on X and the units on Why are spaces number of spaces and the units on X are dollars charge per space. Okay, so what does this mean? Because it's negative. It's going down. And that tells us that he he would rent for fewer spaces for every dollar. He increased the rent charge. So four fewer spaces are rented for every dollar increase and rent. Okay, Now, how about the Why intercept? So the Y intercept? We already looked at a little bit. It's the 0.0 200. And that would be the point where the charge per space was nothing. If he charges nothing, he will rent 200 spaces. Perhaps that means he has 200 spaces. How about the X intercept? That was the 0.50 0 we looked at that a moment ago is well that would be if he charges $50 per space, he won't rent any of them. Charge $50 rent zero spaces.