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Graphing logarithmic functions (example 2)

Video transcript

this is a screenshot from an exercise on Khan Academy it says the inter graphic the interactive graph below contains the graph of y is equal to log base 2 of X as a dashed curve and you can see it down there is that dash curve with the points 1 comma 0 and 2 comma 1 highlighted it just the movable graph to draw y is equal to 4 times log base 2 of x plus 6 minus 7 and so if you happen to have this exercise in front of you I encourage you to do that or if you're just thinking about it in your head think about how you would approach this and I'll give you a hint to go from our original y is equal to log base 2 of X to all of this it's really going to be a series of transformations and on this tool right over here what we can do is we can move this vertical asymptote around so that's one thing we can move and then we can also move two of these points so where we're starting is right we are starting right over there and so let's see and that was just the graph of y is equal to log base two of X so let's just do these transformations one at a time so the first thing I am going to do instead of just doing log base two of X let's do log base 2 of X plus 6 so if you replace your X with an X plus 6 what is it going to do well it's going to shift everything six to the left and if that doesn't make intuitive sense to you I encourage you to watch some of the introductory videos on shifting transformations so everything is going to shift 6 to the left so this vertical asymptote is going to shift 6 to the left it's going to go B instead of being it at x equals 0 it's going to go all the way to x equals negative 6 this point right over here which was at 1 comma 0 it's going to go 6 to the left 1 2 3 4 5 6 and this point which was at 2 comma 1 it's going to go 6 to the left 1 2 3 4 5 & 6 so so far what we have graphed is log base 2 of X plus 6 so the next thing we might want to do is what is 4 times log base 2 of X plus 6 and why you think about it is whatever Y value we were getting before we're now going to get 4 times that so when X is equal to negative 5 we're getting a y-value of 0 but 4 times you are still 0 so that point will stay the same but when X is equal to negative 4 we're getting a y-value of 1 but now that's going to be 4 times higher because we're putting that 4 out front so instead of being at 4 it's instead of being at 1 it's going to be at 4 so this right over here is the graph of y is equal to log base 2 of x plus 6 and then the last thing we have to consider is well we're going to take all that and then we're going to subtract 7 to get to our target graph so whatever points we are here we are now going to subtract 7 so this is at y equals 0 but now we're going to subtract 7 so we're going to go down 1 2 3 4 5 6 7 I went off the screen a little bit but let me see if I can scroll down a little bit so that you can see that see almost there you go now you can see I move this down from 0 to negative 7 and then this one I have to move down 7 1 2 3 4 5 6 & 7 and we're done there you have it that is the graph of y is equal to 4 times log base 2 of X plus 6 minus 7 and we are done