Subscribe! http://www.freemathvideos.com 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 exponential equations when we have multiple transformations. So, in the last video, we just talked about equations that are in the form of y equals bTx or y equals a times bTx. And, basically, we were just kind of showing the different, the y-intercepts, the graphs, and basically just dealt with reflections. Either the reflection of the x-axis or reflection of the y-axis. But, in general, just sketching the graphs. We didn't really get much change with the demand range. It was almost all the same for every single problem.
But now we're going to get into some transformations which actually include some translations. Meaning we're going to be shifting the graph up or down. Now we're still going to have, let's see, we're still going to have some reflections. You can see I have some negatives on the inside and the outside.
And, remember, when we multiply a function by a negative on the outside, that's going to be reflected in the x-axis, and when we multiply by negative inside the function, that's going to be reflecting the y-axis.
Now, as far as translations go, we basically are now adding an h and a k. Now it's very important. h and the k is what we did for quadratics. Same thing that we did for absolute value, radical expressions, whatever types of functions you've already witnessed. h is being subtracted inside the function and k is being added outside the function.
Now it really doesn't matter if you're subtracting and adding, because we're going to be doing, we're going to be doing both, inside the functions. But remember, h is going to be a translation, left or right, and k is going to be a translation, up or down. Or, basically, we're going to be shifting the graph up or down based on the value of k. And we're going to be shifting the graph left or right based on the value of h.
Last thing I want to remember is, we go through this every single time, I don't want to spend a large amount of time explaining it, but it's usually one of the points where students get most mixed up is, remember, it's x opposite of h. OK? So you can always think about that as x minus h. OK?
So h is going to be your shifting left or right. But if it's x minus three, well that's really x minus three. So, therefore, h is equal to three. That means I'm going to shift the graph three inches to the right. Everybody gets confused because they think, oh, it's x minus three. That's going three to the left.
Well, actually, three is positive because the formula is x minus h, so, therefore, it's going to be the opposite. Whereas, if k is positive, you go up. If k is negative, you go down. But we do this for all the functions. I just thought I'd kind of re-hash it again.
Last thing I just kind of want to go through with you is the paragraph. Because, again, I don't have graph paper. The best way to really understand how graphs change based on the base. See here, the base is two . Here, the base is four. Here, the base is e. Here the base is 1/2.
All these different bases are going to affect the graph. I don't have graph paper. I'm not going to spend the time doing a perfect graph. I'm just going to sketch it and all I ask from my students is to sketch it, as well as just provide one point.
And the best point to provide is when you have, if you remember from our last video, the paragraph looks something like this when b is greater than one. So the graph is always going to cross at a. If your equation is y equals b to the x, then the graph crosses at 0 comma b. OK? And then remember if b is less than 1, if b is less than 1, then the graph is going to look something like that.
And also remember we have this asymptote. Now that we have this horizontal asymptote, when we shift left or right, we're going to have to shift that asymptote. And that's very important because that's going to now effect our range of our graph and I'll kind of explain it as we get into it. OK.
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