Hey, everyone. We now know how to graph basic exponential functions of the form f(x)=bx, where b is just some number like 2 or 4 or 12 or whatever. But what if we're given a more complicated exponential function like this g(x) that we have here? How are we going to graph that? Well, you may be worried that we're going to have to take a bunch of different values for x and plug them in and do some messy calculations. But here, I'm going to show you that we're not going to have to do a single calculation because we can simply apply rules of transformations to our parent function bx that we already know how to graph. So all we're going to be doing here is taking the graph of a function that we already know how to get and picking it up and moving it around in order to get the graph of our new, more complicated function. So let's go ahead and get started here.
Now when working with this g(x) here, we see that it looks really similar to 2x but just with a couple more things added in. So we have this negative one and this negative four, both of which represent transformations. Now when working with exponential functions, we're going to be focused on 2 types of transformations in particular, reflections and shifts, because they are the most commonly occurring transformations when working with exponential functions. So just as a reminder, whenever we have a negative on the outside of our function, that represents a reflection over the x-axis, whereas a negative on the inside of our function represents a reflection over the y-axis.
Now when we're working with h and k, this h here represents a horizontal shift. Remember, h horizontal. And then k, of course, represents a vertical shift by some number of k units. So let's go ahead and start graphing our function g(x) here starting with our step 0. Now step 0 is actually to identify and graph our parent function. And this step is really important because we always, always want to graph our parent function first when transforming exponential functions.
So let's go ahead and take a look at our g(x) here. I said that it looks really similar to 2x, and that represents our parent function. So our parent function here is f(x)=2x. And let's go ahead and plot this using a couple of different points. So the first point that we have is going to be negative one, one over b. Now b is the base of our parent function, which here is 2. So my first point will be at negative one, one over 2.
Then my second point will be at 1, which will be the same no matter what your parent function is. And then finally, I'm going to plot one final point at 1b, which remember b is the base, so in this case, 2. So let's go ahead and plot these three points on our graph. So first, negative 1, 1-half, then 0, 1, and finally 1, 2. Now we wanna go ahead and connect all of these points using the shape of our exponential graph. And then we want to do one more thing to get that parent function completely graphed, and that is plot our horizontal asymptote at y=0. That asymptote will always be at y=0 no matter what your parent function is. So we're using a dash line here because it is an asymptote.
Now that we have that parent function graphed, we can go ahead and move on to actually graphing our new transformed function, g(x). So looking at that first step, step 1, we want to go ahead and shift our horizontal asymptote that we just drew. So looking at my function g(x), I'm going to want to shift my horizontal asymptote to y=k. Now when looking at a transform function, I know that k is just added to the end of my function. And here I have this negative 4, so I know that my k value is going to be negative 4. So let's go ahead and shift that horizontal asymptote all the way down to negative 4, of course, still using a dash line because it is an asymptote.
So with step 1 done, we have our horizontal asymptote plotted. Let's move on to step 2 and determine if there is a reflection happening. Now remember, a reflection happens when we have a negative on the inside or the outside of our function. And looking at my function, g(x), it looks like I did not have a negative get inserted into my new function, so I don't have to worry about a reflection. And I can move on to the second part of step 2, part b, to shift our test points by h and k. Now we're going to go ahead and identify h and k. We know that k is already negative 4 because we just moved our horizontal asymptote. And then for h, I have this x-1 here. Remember, when dealing with transformations, it's x-h. So here, my h value is simply 1.
Now we're going to go ahead and take those points from our parent function, our test points, and shift them 1 over to the right and 4 units down in order to get the points of our new transformed function. So starting with this first point here, I'm going to go one over and then 1, 2, 3, 4 down to get my new point. Then for my second point, one over again And then 1, 2, 3, 4 down to get my new point. And then my last point, 1 over. And then 1, 2, 3, 4 down to get my new point. So I have all of my new transformed points from my original function. And my very last step, my step 3 here, is just to go ahead and connect everything and, of course, approach my asymptote.
So here I can go ahead and connect my points and then approach my asymptote on this side. So here we can see that our original function just got picked up. It got shifted over and down a little bit in order to get our new function. So now that we have fully transformed our function, we have our graph of g(x), we can go ahead and identify our domain and our range. Now remember, the domain of any exponential function is always going to be the same. It's always going to go from a negative infinity to infinity or simply all real numbers.
Now for our range, this depends on our placement of our graph regarding our asymptote. So here, if our graph is above our asymptote, which here it is, we can go ahead and state that our range goes from k, that value of k, to infinity. Now here my value of k is negative 4, so I know that all of my y values are going to start at negative 4 and go to infinity. So here it is from k to infinity negative 4. Now because this one is above, that's what our range is. But if it happened to be below, our range would instead go from negative infinity to k, just depending on what's happening with our graph.
Now that we know how to transform exponential functions, let's get some more practice. Thanks so much for watching.