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Interactive video lesson plan for: Master Graphing Logarithmic Equations with multiple transformations

Activity overview:

Subscribe! Want more math video lessons? Visit my website to view all of my math videos organized by course, chapter and section. The purpose of posting my free video tutorials is to not only help students but allow teachers the resources to flip their classrooms and allow more time for teaching within the classroom. Please feel free to share my resources with those in need or let me know if you need any additional help with your math studies. I am a true educator and here to help you out.Welcome, ladies and gentlemen. So what I'd like to do is show you how to graph logarithmic equation with transformations, other than just reflections and dilations, stretching and compressing. So basically what we're going to be doing is we added back our two variables, h and k. Which again, remember, h is going to be shifting your graph left or right. k is going to be shifting your graph up or down.

Please also remember, though, that we can write-- where is my blue? I lost-- there's my blue. Please also remember that we can write x minus h like this, x minus h and put that h in parentheses because h is technically in the positive. So whenever doing this, that h is always kind of like in the opposite direction, which confuses students all the time. And even though I've gone over it in different videos, I wanted to just kind of highlight it again.

Let's see, do I have at least a-- OK, so, I'll kind of take a stab here at this one and a kind of explaining it but then I'll kind of move forward with it a little bit quicker for the rest of videos because I don't want to be redundant for those of you that understand it. But basically when we're grappling multiple transformations, the main important thing we want to do is graph our paragraph. Now, basically we're always going to have-- our parent graph is always going to have an x-intercept as long as the coefficient of our x is 1, which we have in case of all of these, except for our final last problem, which I again will explain at the end.

But since all of the rest of our variables, our x has a coefficient of 1 or negative 1, that's just a reflection, we know that the paragraph is going to look somewhat like this, 1 comma 0 and goes like that with a nice little vertical asymptote. So basically what we're going to do is we're going to take that graph and we're going to transform it based on the transformations or reflections. So we're going to-- oh, did I choose all-- oh, there's a couple different bases.

And remember, like numbers in front or the bases. Those are going to affect the graph. The graph could either be stretched vertically. The graph could be compressed vertically. And that's all dependent on the bases or the number that's multiplying by a logarithm. But again, I'm not going to be concerned about the perfect slope, you might say, of the graph. I just want to kind of get the main version of the graph correct as well as apply the operations.

You can create table or you can use a graphing calculator to kind of see how perfect your graph is compared to what the actual answer is. However, they're all going to look pretty close to the same. OK, so let's get into y equals log base 5 of x minus 1. Remember, I stated that this can be written as x minus h. OK, so hopefully you see that I can put parentheses around here, this one. So it's x minus h. x minus 1. x minus h. x minus 1.

So therefore, h is equal to a positive 1. What that's going to tell me to do is I'm going to be shifting the graph to the right 1 unit. Since the only thing I have is a base here of 5, which is just going to be affecting kind of my transformation of the graph, I'm just going to sketch my normal parent graph. And then I know I'm going to be shifting my graph one unit to the right, so that's going to move over there.

Now, what's very important about transformations, especially horizontal transformations with logarithms, remember, we have this asymptote here. Well, if I'm shifting the graph one unit to the right, that means my asymptote has to be shifted one unit to the right. So that's very important because now what that's going to do is that's going change our domain and range. The domain is a set of all x values. How far left does the graph go? How far to the right does the graph go?

Well, you can see in our typical standard form-- in our [INAUDIBLE] graph, the asymptote was at 0. So the farther left the graph went was at 0. Well, now the farthest left graph goes is positive 1. And the farthest it goes to the right is infinity. So the domain is 1 comma 0. The range still goes infinitely down, infinitely up, so that's going to be from negative infinity to infinity. All right, so now we have ln of x. And again, ln of x is just a logarithm with base e.

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