Hey, everyone. We just learned how to graph basic log functions using the fact that they're the inverse of an exponential function. But what if we have to graph a more complicated log function like this g of x we have here? We might be worried that this is where it's going to start to get more challenging and switch up from exponential functions, but you don't have to worry about that at all because we're going to do the same thing we've done a million times before and not do any calculations, just some transformations. So let's go ahead and jump right in. Now our function g of x here looks really similar to log2(x), but just has a couple of extra things going on here, this one and this four, both of which represent transformations.
Now, just as a quick recap of the most common transformations, remember that if you have a negative on the outside of your function, that represents a reflection over the x-axis, whereas the negative on the inside of your function represents a reflection over the y-axis. Then, of course, we have h, which represents a horizontal shift, and k, which represents a vertical shift. Now, just as we did for exponential functions, when working with log functions, we are always going to want to graph our parent function first because the base of your log function isn't always going to be the same. So let's go ahead and do that for this function g of x that we have here and start with step 0 and plot that parent function first.
So here, we want to go ahead and identify our parent function. I mentioned that in g of x, it looks really similar to log2(x) because that actually is our parent function here. So our parent function f of x is equal to log2(x). We want to go ahead and graph three points here at 1 over b where b is our base, so in this case 1/2, -1. Then our second point will be at 10, which will be the same no matter what the base of your log function is. And then our last point will be at b. base 1. So b here is 2. So that last point is at 2, 1. Now, these are going to be the easiest points for us to plot. And they're going to serve as sort of test points as we graph our new function. So plotting those on my graph, I have one, two, -1 then 10, and then 2, 1. Now I can go ahead and connect those points to get my graph here. And then I can also go ahead and plot my vertical asymptote at x equals 0 in order to get my entire parent function all ready to go and ready to be transformed.
Now that we have that parent function, we can go ahead and start plotting our actual function g of x here starting with step 1 and shifting that vertical asymptote to x equals h. Now here looking at my function I have that h is this one because remember x - h. So here h is just positive one. So I can go ahead and shift my vertical asymptote right over to that one using, of course, a dash line because it is an asymptote. K. Now that we have finished step 1, we can move on to step 2 and identify whether or not there's a reflection happening. Now remember, reflection happens if there is a negative on the inside or outside of our function. And here, looking at my function g, I don't have a negative that got added, so I don't have to worry about that reflection. And I can go ahead and move on to the second part of step 2 and shift my test points by h and k. Now we already identified h as being positive 1 and looking over at my function here this negative 4 tacked on the end here represents k. So I'm going to shift my test points of my function here by 1 point to the right and 4 points down. So let's go ahead and do that starting with this first point. I'm going to go 1 to the right and 1, 2, 3, 4 down, ending up right here. Then my next point, 1 to the right and 1, 2, 3, 4 down, and my final point, 1 over to the right, and 1, 2, 3, 4 down. So I have my new points here shifted by h and k, and then I can go ahead and move on to the very last step in plotting my function, which is going to be to sketch my curve approaching the asymptote.
So here I can go ahead and connect all of my new points and then approach that asymptote on the bottom there. Now that we have our complete picture of our graph, we can go ahead and identify our domain and our range as well. Now remember, our range is always going to be the same for our log functions no matter what that base is or what other shifts and transformations are happening. It's always going to be all real numbers. So there's my range, all real numbers. And then, of course, my domain actually depends on my asymptote and which direction I'm approaching my asymptote from. So looking at my graph here, I'm approaching my asymptote from that right side. And when that happens, that means that my domain is going to go from my asymptote at h to infinity. Now here h is 1, so my domain is going to be from 1 to infinity. Now if my graph was approaching my asymptote from the left side instead over here, my domain would then go from negative infinity until I reach that asymptote at h. Now that we know how to graph log functions using transformations, let's get some more practice.