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CCSS.Math: ,

- [Instructor] This is a screenshot from an exercise on Khan Academy, and it says the intergraphic,
the interactive graph below contains the graph of y is equal to log base two of x as a dashed curve, and you can see it down
there as a dashed curve, with the points one comma zero and two comma one highlighted. Adjust the movable
graph to draw y is equal to four times log base two
of x plus six minus seven. 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 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
two 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 two of x plus six. So if you replace your
x with an x plus six, 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 six to the left. So this vertical asymptote is
going to shift six to the left it's gonna be, instead of
being at x equals zero, it's going to go all the way
to x equals negative six. This point right over here,
which was at one comma zero, it's going to go six to the left, one, two, three, four, five, six. And this point, which as at two comma one, is gonna go six to the left, one, two, three, four, five, and six. So so far what we have graphed is log base two of x plus six. So the next thing we might wanna do is what is four time log
base two of x plus six. And I want you to think about it is whatever y-value we were getting before, we're now going to get four times that. So when x is equal to negative five, we're getting a y-value of zero, but four times zero is still zero, so that point will stay the same. But when x is equal to negative four, we're getting a y-value of one, but now that's going to
be four times higher, 'cause we're putting that four out front, so instead of being at four, instead of being at one
it's going to be at four. So this right over here is the graph of y is equal to log
base two of x plus six. And then the last thing
we have to consider is well we're gonna take all of that and then we're going to subtract seven to get to our target graph. So whatever points we are here, we are now going to subtract seven. So this is at y equals zero, but now we're going to subtract
seven, so we're going to go down one, two, three,
four, five, six, seven, 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, almost, there you go, now you can see. I moved this down from
zero to negative seven, and then this one I
have to move down seven, one, two, three, four,
five, six, and seven, and we're done, there you have it. That is the graph of y is equal
to four times log base two of x plus six minus
seven, and we are done.