 Of all the functions, in all of the applications in the entire universe, the type we like the best are linear functions. A linear function is a function of the form f of x equals ax plus b, where a and b are real numbers. So for example, c of x equals 50x plus 1500, or r of x equals 517s x minus 4000, or we could change variables, h of t is 150t minus 400. So let's play around with these linear functions. Suppose the cost c of x to produce x units of a product is modeled by c of x equals 500x plus $1000. Let's find and interpret c of 100. So remember, equals means replaceable, and also paper is cheap, write things down. So if we want to interpret this, it's helpful to write things down. And so here we have x, well that's the units of a product. So we might say that x is the same as the units produced. c of x, on the other hand, well that's the cost to produce x units. But we also know that c of x is equal to 500x plus $1000. Now we want to say something about c of 100. So let's write that down. And the first thing to recognize here is that we replaced x with 100. And remember, equals means replaceable, and so that means every time we see an x, we can replace it with a 100. So let's replace this x with 100, and everything else stays the same, so 100 is equal to the units produced. But wait, there's more. There's another x we can replace with 100. And that's this one. So we'll replace it, and we get an expression whose value we can compute, which works out to be $51,000. And so we can put it together. c of 100 is equal to $51,000, and what this means is that 100 units produced at a cost of $51,000. Now part of the reason that we like linear functions is that if f of x is a linear function, then the graph of y equals f of x will be a straight line with a slope and y intercept. So if c of x equals 50x plus 1500, then the graph of y equals c of x will be, well, equals means replaceable. c of x is equal to 50x plus 1500, so every time we see c of x, we can replace it with 50x plus 1500. So here's a c of x, we'll replace it, and so we'll have y equals 50x plus 1500. And this is the equation of a straight line, and so it's useful to keep in mind the graph of y equals mx plus b is a straight line with slope m and y intercept 0b. So this graph will have slope m equal to 50 and a y intercept of 01500. So for example, let's go back to our cost function, c of x equals 500x plus 1000. Let's find and interpret the slope and the y intercept. Well, finding the slope and the y intercept is easy. The graph of c of x will be y equals 500x plus 1000, which will have slope m equal to 500 and y intercept 01000. So these are the values, what do they tell us? Well, remember that the slope is rise over run, and the easiest way to view any slope m is as m over 1. So our slope m is 500. So if m equals 500, which is to say 500 over 1, then we have the following. The y values. Well, y is c of x, that's our cost, have a rise of $500. Whenever the x values, those are the units produced, increase by 1. And so we might interpret this as the cost of production increases by $500 for every additional unit produced. This is sometimes referred to as the marginal cost. What about our y intercept? Remember the y intercept is a point with x equal to 0. So if the y intercept is 01000, this means that when x is equal to 0, well, x is the number of units produced. So when x equals 0 items are produced, our y value is 1000. Well, our y value is c of x, and c of x is our cost. So the cost to produce them is $1000. And so the way we can read this is that even if 0 items are produced, even if no items are produced, it will still cost $1000. And this is sometimes referred to as the fixed cost. This is what it's going to cost you to do absolutely nothing.