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.