In this problem, we want to graph our given function g(x) as a transformation of f(x) which is equal to 12x. The function I have g(x) is -12x+3. The first thing that you might notice here is that we have a negative sign. And I know that I've said before that the base of an exponential function cannot be negative, and this does not violate that. So I have this negative sign outside of these parentheses. If this was -12 to the power of x, that's when we would have a problem. So I just wanted to take a second to address that. It's totally okay if your negative is on the outside of your base as long as it's not being raised to that power because here we're taking one half, raising it to the power of x, and then making it negative, and that's what makes this okay. So let's go ahead and get right into graphing, starting with step 0.
Step 0 is to identify and graph our parent function, f(x) = b to the power of x. We know here that our parent function is 12x. So, going ahead and plotting the points that I need, I want to graph negative 1 over b which is just negative one two, 1, and then 1 over b, where b here is one half. So let's go ahead and plot these three points: (-12, 0), (1, 1), and (1, 1/2). Now we can go ahead and connect all of these. And then, of course, we want to plot our horizontal asymptote at y = 0, which is just at our x-axis using a dashed line. So we have our parent function here.
Let's go ahead and get into our new function and transform this parent function. Our new function that we have is -12x+3. The very first thing I want to do is shift my horizontal asymptote to y = k. Here, I have this plus 3 on the end, which is my value for k. So I know that I need to have my horizontal asymptote at y = 3. Let's go ahead and put our horizontal asymptote using a dashed line on my graph up here at y = 3.
Now that we're done with step 1, we can move on to step 2 and decide if there is a reflection happening. Remember, reflection happens if we have a negative outside or inside of our function, and we do have this negative added to the outside of our function. So that tells us that, yes, we do have a reflection, which means that we need to take those test points from our parent function and reflect them here over the x-axis because that negative is outside of our function. So let's go ahead and do that here. Taking my points, so starting with this point, I'm going to reflect that over the x-axis. So my new point will end up here. Same thing for this one here, going to end up right here. And then my last point right here. So I have reflected my new points, but now I need to shift them.
Remember, we shift them by h, k. Here, we don't have a value for h, it's just 0, so I don't have to worry about any horizontal shift. But I do have a vertical shift by k. So we need to take all of these points and move them up by 3. So let's go ahead and do that. Shift up by 3, shifting. And then my other point here by 3, and then my last point up by 3 as well. So now I have my final points that I can go ahead and sketch and connect these points using a curve, of course, approaching my asymptote. So here we're approaching that asymptote and then on this side as well. So we have our new function here. We have completed graphing it, but let's go ahead and identify this additional information, our domain and range.
Remember, our domain will always be the same, no matter what. It's always going to be all real numbers, so I don't have to decide anything or figure anything out there. But for my range, I need to look at whether my function is above or below the asymptote. Now, since my asymptote is right here and my function is down here, that tells me that it is below my asymptote. So my range is going to go from negative infinity until we reach that asymptote at k, which in this case is at 3. So here, my range is from negative infinity to 3. Now that we've identified all of that information, let's get into some practice.