 In this video, I wanted to provide some more practice at simplifying difference quotients. That is, this thing right here, it's a quotient of a difference, very clever name, ha ha ha. Anyways, it's essentially just the average rate change formula. So given the function f of x equals two x squared minus five x plus one, how can we simplify this difference quotient, right? Remembering along the way that h doesn't equal zero because oh no, we can't divide by zero. We don't want that to happen. So as we try to compute this thing, notice that f of a plus h means plug a plus h into the function. So each x gets replaced with an a plus h. This would look like for us, two times a plus h squared minus five times a plus h plus one. And then we're gonna subtract from that f of a, which just means you plug in a for each of these x's here, in which case you end up with two a squared minus five a plus one. And this all sits on top of h. Now, our motivation when we simplify a difference quotient is our goal is to get rid of the h on the bottom. H stands for hate. We gotta get rid of the hate. And that is what our goal is trying to be right now. And that'll come by simplifying or expanding the numerator in some degree, right? We wanna cancel some like terms. For example, there's a one minus one, those cancel out pretty quickly. The next thing I wanna do is foil out this a plus h squared. If we do that, we're gonna get two times a squared plus two a h plus h squared, just by the usual foil method, minus five times a plus h. And then we subtract from here two a squared minus five a. And this sits above h. The next thing I wanna do is distribute all of these coefficients through, oops, I missed one, distribute, distribute. And if we do that, we're gonna end up with a two a squared plus a four a h plus a two h squared. We're gonna get a minus five a and we're going to get a minus five h. And then we subtract from that a two a squared minus five a. Now that everything in the numerator is completely expanded out, don't forget your denominator. Of course, h, it's a long for the ride. When you multiply the denominator out, you'll notice that everything in the second group is gonna cancel with something in the first group. It's sort of like playing hide and go seek right now, right? You'll notice here there's a two a squared and a two a squared. They're gonna cancel each other because after all, there is a negative sign right there. You're also gonna notice that there is a negative five a right here and there's a negative five here. They're gonna cancel each other because again, we're subtracting them. They cancel each other out. You're gonna notice that oftentimes, college algebra very much is a cancel culture. We're always trying to cancel everyone, all the variables, they must be canceled. When we look at who survives the massacre, which we call the numerator, you're gonna have a four a h plus a two h squared minus a five h in the numerator. This is all over h of course. But notice actually a happy accident has now occurred. There is a multiple of a for every term in the numerator. If we factor out that a, not a, I'm sorry, if we factor out the h, that will leave behind a four a, that's where the a came from, plus a two h minus a five times that eight by h over h. You'll notice that now, because we have a multiple of h in the numerator, that'll cancel with the h in the denominator we never wanted in the first place. And that's left leaving us with a four a plus two h minus five. And this is the simplified difference quotient. This gives us a simple formula for the average rate of change. Now, if this was a calculus class, we would take the next step and send h to zero, right? But no, no, no, no, this is not a calculus class. We don't do that. That's really all the difference from calculus, right? Calculus is just going the next step. We basically have to sit on the precipice of this cliff watching, watching, watching without the capacity of, you know, paragliding off of it, paragliding off of that. That's what we wanna do. The fun part's next, but we can't do it because our curriculum restricts us from taking the next step, from jumping off that cliff into the beautiful water below. When we get this average rate of change, we're just gonna be four a plus two h minus five.