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# Graphing logarithmic functions (example 1)

CCSS.Math:

## Video transcript

we are told the graph of y is equal to log base 2 of X is shown below and I say graph y is equal to 2 log base 2 of negative X minus 3 so pause this video and have a go at it and the way to think about it is is that this second equation that we want to graph is really based on this first equation through a series of transformations so encourage you take some graph paper out and sketch how those transformations would affect our original graph to get to where we need to go all right now let's do this together so what we already have graphed I'll just write it in purple is y is equal to log base 2 of X now the difference between what I just wrote in purple and where we want to go is in the first case we don't have multiply anything times our log base 2 of X while in our end goal we multiply by 2 and our first situation we just have log base 2 of X while in here we have log base 2 of negative X minus 3 and in fact we could even view that as it's the negative of X plus 3 so what we could do is try to keep changing this equation and that's going to transform its graph until we get to our goal so maybe the first thing we might want to do is let's replace our X with a negative x so let's try to graph y is equal to log base 2 of negative x and other videos we've talked about what transformation would go on there but we can Intuit through it as well now whatever value Y would have taken on at a given x value so for example when x equals 4 log base 2 of 4 is 2 now that will happen at negative 4 so log base 2 of the negative of negative 4 well that's the log base 2 of 4 so that's still going to be 2 and if you were to put in let's say a whatever was happening at one before log base 2 of 1 is 0 but now that's going to happen at negative 1 cuz you take the negative of negative 1 you're gonna get a 1 over here so log base 2 of 1 is zero and so similarly and when you had at x equals eight you got two three now that's going to happen at x equals negative eight we are going to be at three and so the graph is going to look something something like what I am graphing right over here all right fair enough now the next thing we might want to do is hey let's replace this X with an X plus three because that'll get us at least it in terms of what we're taking the log of pretty close to our original equation so now let's think about y is equal to log base two of and actually I should put parentheses in that previous one just so it's clear so log base two of not just the negative of X but we're going to replace X with X plus three now what happens if you replace X with an X plus three or you could even view X plus three is the same thing as X minus negative three well we've seen in multiple examples that when you replace X with an X plus three that will shift your entire graph three to the left so this shifts shifts three to the left if it was an X minus three in here you would shift three to the right so how do we shift three to the left well what when the points where we used to hit zero are not going to happen three to the left of that so we used to hit it at negative at x equals negative one now it's going to happen at x equals negative four the point at which Y is equal to 2 instead of happening at x equals negative four is not going to happen three to the left of that which is x equals negative seven so it's going to be all right over there and so and the point at which the graph goes down to infinity that was happening as X approaches zero now that's going to happen as X approaches three to the left of that as X approaches negative three so I could draw a little dotted line right over here to show that as X approaches that our graph is going to approach zero so our grass gonna look something something like like this like this and this is all hand-drawn so it's not perfectly drawn but we're awfully close now to get from where we are to our goal we just have to multiply the right-hand side by two so now let's graph why not two let's graph y is equal to two log base two of negative of X plus three which is the exact same goal as we had before I've just factored out the negative to help with our transformations so all that means is whatever Y value were taking on and to give an X you're not going to take on twice that Y value so where were you were at zero you're still going to be zero but where you were to you were now going to be equal to four and so the graph is going to look something something like what I am drawing right now and we're done that's our sketch of the graph of all of this business and once again if you're doing it on Khan Academy there would be a choice that looks like this and you would hopefully pick that one