Graphing logarithmic functions (example 1) | Algebra 2 | Khan Academy

We're 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. The way to think about it is that this second equation that we want to graph is really based on this first equation through a series of transformations.

So I encourage you to take some graph paper out and sketch how those transformations would affect our original graph to get to where we need to go. Alright, now let's do this together.

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 multiply anything times our log base 2 of x, while in our end goal, we multiply by 2. In our first situation, we just have log base 2 of x, while in here we have log base 2 of negative x minus 3.

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.

Let's try to graph y is equal to log base 2 of negative x. In 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 still 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 1 before, log base 2 of 1 is 0, but now that's going to happen at negative 1 because you take the negative of negative 1, you're going to get a 1 over here.

So log base 2 of 1 is 0. And so similarly, when you had at x equals 8, you got to 3; now that's going to happen at x equals negative 8. We are going to be at 3. And so the graph is going to look something like what I am graphing right over here.

Alright, fair enough. Now, the next thing we might want to do is, hey, let's replace this x with an x plus 3, because that'll get us, at least 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 2 of, and actually, I should put parentheses in that previous one just so it's clear, so log base 2 of not just the negative of x, but we're going to replace x with x plus 3.

Now, what happens if you replace x with an x plus 3, or you could even view x plus 3 as the same thing as x minus negative 3? Well, we’ve seen in multiple examples that when you replace x with an x plus 3, that will shift your entire graph 3 to the left.

So this shifts 3 to the left. If it was an x minus 3 in here, you would shift 3 to the right. So how do we shift 3 to the left? Well, when the points where we used to hit 0 are now going to happen 3 to the left of that. So we used to hit it at negative at x equals negative 1; now it's going to happen at x equals negative 4.

The point at which y is equal to 2 instead of happening at x equals negative 4 is now going to happen 3 to the left of that, which is x equals negative 7. So it's going to be right over there. And the point at which the graph goes down to infinity that was happening as x approaches 0, now that's going to happen as x approaches 3 to the left of that, as x approaches negative 3.

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 graph's going to look something 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 y not two; let's graph y is equal to two log base 2 of negative of x plus 3, 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 we're taking on at a given x, you're not going to take on twice that y value. So where you were zero, you're still going to be zero, but where you were two, you're now going to be equal to four. And so the graph is going to look 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.