WEBVTT mathematics/pre-calculus/selhorst-jones
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Hi--welcome back to Educator.com.
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Today, we are going to talk about finding points of interest.
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We often want to find the interesting points on a graph, places where something special occurs.
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A graphing calculator will allow us to easily find these locations.
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You can use a graphing calculator to find things like roots or zeroes (the same thing, just different names),
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the locations where our function is equal to 0; you can use it to find relative minimums
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(sometimes just called min), which is the lowest local value in an area, relative maximums
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(sometimes just called max), the highest local value in an area,
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and intersections (the location where two functions intersect each other).
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The specific menu to get access to these varies from one calculator to another.
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But the menu choices should look something like these things right here: zero, min, max, intersection.
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If you can find a section like that, you have found the part where you can get that information out of your calculator.
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You set up your function, then choose whichever one you want to use.
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It is important to note that all of these are usually solved numerically.
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Some calculators can solve them precisely, but most graphing calculators will solve them numerically.
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That is to say, you will get answers that are accurate up to quite a few decimals--you will be able to get
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four or five decimals' worth of accuracy, maybe even more.
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But they won't necessarily be perfectly precise; it is like the difference between saying π and saying 3.1415.
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3.1415 is a very good approximation of π, but there is still more precision in the actual number of π.
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So, the answer that you get from your calculator, when you use things for roots, zeroes, relative minimums/maximums,
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intersections, and things like that will be good answers; but they won't necessarily be perfect answers.
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They will be approximations; and that is something you want to keep in mind sometimes.
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So, how do we actually find a point of interest with a graphing calculator?
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For most points of interest, the process is going to go like this: you start by graphing the function.
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Now, in this specific case, I actually graphed a parabola; and I knew that because I knew I had x² at the front of it, and then some other stuff.
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But when I graphed it, I see that I get this on my graphing calculator screen.
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So instead, what I want to do is: I might (this part is optional, but I might) want to resize my viewing window so that I can see more of what is going on.
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So, I can adjust the viewing window, and I see that it is a parabola.
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And so, I have a bunch more stuff going on here; it is not just a straight line that I am working with--I am working with a parabola.
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So, you can adjust your viewing window.
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Depending on the situation, you might not need to adjust the viewing window.
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For example, if you are looking for a zero at some specific area (it is going to be between an x interval of maybe 2 and 5),
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you don't really care about a window any farther than going from an x interval of 2 to 5,
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because that is all you are looking for; and you know that your y's really only have to be near 0, because you are looking for a root.
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But depending on the situation, it can be useful to see the entire thing, or at least a good sense of how the function works, before moving on.
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Next, you choose the point of interest type: what you are specifically looking for.
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In this case, let's say we are looking for a root, and we want to find out what this root is--that one right there.
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We go to the menu; we choose "zero."
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At that point, you identify where the search should occur.
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This won't be necessarily on all graphing calculators; but many graphing calculators will require you to put an interval:
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what is the lowest place that we can look from and the highest point that we are looking to?
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What is the lowest x-value that we have to look from and the highest x-value that we are going to work up to?
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And then, it will search within that interval.
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So, we start, and we say, "We know that it is going to be above here and below here for this x-interval,
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so we are just looking somewhere inside of this interval."
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Next, you give your graphing calculator a guess.
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You don't necessarily have to have your guess be perfect; it doesn't have to be right on top of the thing you are going for.
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But the closer it is to where you are going, the faster your graphing calculator will be able to figure it out.
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The algorithms that it is using to actually figure this out...it just starts from somewhere that you tell it to start, and then it works out either way, effectively.
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So, it works out from some starting place; so if your starting place is closer to where you are going, it will make it a faster process.
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We choose some guess somewhere, and then it cranks through it, and it gets us an answer.
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We have a result, and it will display it somewhere on your graph.
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It might display it next to the point; it might display it at the bottom of the window or the top; it depends on your specific calculator.
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But it will punch out some value; and so, in this case, we managed to get that it happens at 2.81; so we have three decimal places of accuracy.
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The process works pretty much just like this for if you are looking for the zero, the minimum, or the maximum.
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But it works a little bit differently if you are looking for the intersection of two functions.
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For that, you graph both functions; so you will have to graph one of the functions,
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and then you choose another; you set another function graphing the other one.
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Then, you will choose "intersect" in your menu (you choose your point of interest type).
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You identify the two functions: instead of identifying where the search occurs, between what locations,
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you say, "Here is the first function I care about; here is the second function I care about."
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And then, you give some guess of about where you think the intersection occurs.
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And then, once again, it runs through some algorithm, and it figures out where the actual intersection is.
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There are some advanced techniques that we can talk about, as well.
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One thing that you might want to do is be able to solve for an arbitrary output value on a function.
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Occasionally, you need to solve some function and find out what input value will give a certain output.
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For example, we might have the function f(x) = x³ - 27x² + 9.
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And we want to find all of the x-values such that f(x) = 419, all of the values that we can plug in and get 419 out of it.
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In other words, what we are trying to do is find all of the solutions to 419 = x³ - 27x² + 9.
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If you are not sure why that is, well, we have f(x) here and f(x) here.
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What we want f(x) to be is 419; so we swap it out here, and we have the equation 419 = x³ - 27x² + 9.
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So, we are effectively just looking to solve this equation.
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What are all the x-values that get this equation solved?
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We can use a graphing calculator to quickly find a close approximation.
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We can write the above as 0 = x³ - 27x² + 9 - 419.
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We are just moving the 419 to the other side of the equation through subtraction.
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Now, at this point, if we have 0 = this stuff, well, what we can do is say, "Let's just make this y = this stuff."
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So, we swap out the 0 for a y; and now we have something that we can graph.
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We can graph y = x³ - 27x² + 9 - 419.
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So, we plug that into our graphing calculator, and we use the calculator to solve for all zeroes,
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because, if we find a zero, if we find each of the zeroes to this equation right here,
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if we find 0 = this equation, well, then we will have satisfied this equation.
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And if we have satisfied that equation, we know that we have satisfied that equation.
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And if we have satisfied that equation, we have figured out all of the places where x equals 419.
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And so, we are done; this is a really great way, if you want to just figure out what input values will cause a function to give certain output values.
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You can work through this method, and you can solve arbitrarily for anything; it is pretty great.
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Another advanced technique to know about is for calculus.
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If you look in the menu that allows you to find the various points of interest,
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you will normally also find some options for calculus.
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The derivative, dy/dx, finds a numerical approximation of the derivative at the point you choose.
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Once again, it is not a precise value; but it is a pretty good numerical approximation.
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And the integral finds a numerical approximation (once again, not the precise value, but a good approximation) of the definite integral for the interval you choose.
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That is all of the area underneath some starting location to some ending location underneath your curve.
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Now, don't worry if these options don't really make sense to you right now.
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That is perfectly fine; this is for a future course when you get into calculus.
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It is some interesting stuff now; but don't worry about it right now.
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You will be learning about these things later on in a calculus course.
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But when you get to calculus, keep them in mind; keep the fact that your calculator has the ability to figure out derivatives; it has the ability to figure out integrals.
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They can be really handy for checking your work or for solving difficult problems when you are in a calculus class.
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They can really help you out in that place.
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Finally, how do you show your work with this stuff?
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The problem with all of these methods, and rightly so, is that you don't really do any solving on your own.
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You are just letting the calculator do all of the work for you.
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While that is OK in some situations; you still should be able to work out solutions for problems like this on your own.
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This means that you should be able to do this without a calculator, too.
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Most teachers won't accept "because my calculator said so" as the work for an answer!
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Most teachers are going to say, "No, you have to be able to solve this on your own, not just rely on a calculator being able to do it."
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And that is perfectly reasonable; the point of being in math class is being able to understand how this stuff works.
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And if we are completely relying on a calculator, it is not that great.
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You don't want to become dependent on your calculator for all of your solving.
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You can use it as a way to check your work or solve a problem that you can't figure out algebraically right now.
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But don't let it become a crutch that replaces thinking.
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You want your calculator to be a tool that helps you do math, but not the only way that you can do math.
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It is a great tool, but if it ends up replacing all of your ability to solve stuff on your own, that is kind of not what you are going for.
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You really want to be able to understand and think, and just appreciate it as a tool.
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But it is a great way to check your work and to give you a head start on being able to figure out a problem where you are not quite sure which direction to go.
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It can give you a hint, because you might be able to figure out the answer before you even work towards it.
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All right, we will see you at Educator.com later--goodbye!