Hey everyone. Now that we know how to fully graph a polynomial function, let's work through this example together and graph this polynomial function, and then determine both the domain and range. The function I have here is f(x) = 3x^3 + 12x^2 + 12x. So jumping into step 1 determining our end behavior, we want to go ahead and look at our leading term here which is 3x^3. Now the very first thing I want to look at is my leading coefficient which in this case is 3, and it is positive. So that tells me that the right side of my graph is going to rise. Now because my degree here is 3, this is an odd number that tells me that the ends are going to have the opposite behavior. So going ahead and sketching that on my graph here, I know my right side is going to rise, and then my left side is going to do the opposite because the ends have the opposite behavior. Remember that this doesn't need to be precise yet, we're just sketching what's eventually going to happen on our graph. So we're done with step 1. Let's move on to step 2 and determine our x intercepts and their behavior. Now for this one we need to do a little bit more work and do some calculations, so let's go ahead and set up our equation f(x) = 0 so that we can solve that. Now I'm gonna take my function here, which is 3x^3 + 12x^2 + 12x and set it equal to 0. Now let's come down here and work this problem out. So looking at this function or this equation that I have here, I see that I can go ahead and factor out a greatest common factor out of each of these terms. Now they all have the common factor of 3x that I can go ahead and pull out. So taking out that factor, I'm left with x^2 + 4x + 4 = 0. Now here I don't have to worry about this 3x anymore, but I can further factor this. And you may recognize that this is a perfect square trinomial. So I can actually factor this into (x + 2)^2 = 0. So now this is fully factored, and I can go ahead and take each factor and set them equal to 0. So I have 3x = 0 and then x + 2 = 0. Now solving for x in each of these, this 3 isn't going to do anything. I'm simply left with x = 0. And then over here, I can subtract 2 from both sides, canceling that out, leaving me with x = -2. So these are my 2 x intercepts, x = 0 and x = -2. Now we need to determine the behavior of the graph at each of these points, so we wanna look at the multiplicity. Now looking at my x = 0, I see that it comes from the factor 3x, which only occurs once. So this has a multiplicity of 1. Now 1 is an odd number, so that tells me that my graph is fully going to cross the x-axis at that point. Now looking at my other factor x = -2, I see that this comes from the factor x + 2 which is squared. So this happens twice. It has a multiplicity of 2. Now because that multiplicity is even, that tells me that my graph is simply going to touch the x-axis and bounce right back off at that point. So we have our x intercepts and their behavior. Let's go ahead and put them on our graph. So looking at x = 0, I know it's going to cross the x-axis at that point, and then x = -2. It is only going to touch and bounce off, which might be shaped something like this. Remember that we're just sketching this right now. We'll connect it all with more information later. So we have our x intercepts. Let's move on to look at our y intercept, which we can find by calculating f(0), plugging 0 into our function. Now we can go ahead and use our factored version here. We have this 3x \times (x + 2)^2, or you can always go back to your original function and plug 0 in there. I'm gonna use the factored version here because I think it's gonna be a little bit easier. So plugging 0 in, I get 3 \times 0 \times (0 + 2)^2. And now all of this is getting multiplied by a 0, so that tells me that I'm simply going to be left with 0 here. So my y intercept is just 0, which I don't need to plot on my graph because I already have it at that origin point there from my x intercept. Now that we have those done, let's move on to determine our intervals. So let's take a look at our graph. Now we wanna go in our graph from left to right looking for points that we know and identifying those intervals of unknown behavior. So from negative infinity until I reach my first known point, I first know something at x = -2. So my very first interval is from negative infinity until I reach -2. Then going from -2 until I reach my next known point, it is at x = 0. So my next interval is simply -2 to 0. And then finally from 0, there is no other known point on my graph, so my last interval is from 0 all the way to infinity. Now that I have my intervals of unknown behavior, I'm gonna go ahead and identify a value for x in each of those intervals that I can calculate a point for. So my first interval from negative infinity to -2, I wanna choose a point that I will be able to plot my graph, which here I'm just gonna go ahead and choose -3. Then from -2 to 0, I'm just gonna go ahead and choose a point right in the middle of that interval at -1. Then finally from 0 to infinity, I'm just gonna see how quickly it increases from there, and I'm gonna go ahead and choose x = 1. Okay. Now that we're here, we wanna take each of these x values and plug them into our function to calculate f(x) to get that ordered pair. So let's come down here and plug all of these into our function. I'm first going to calculate f(-3) by plugging -3 in for x to my function. Now again, you can use your factored form or you can use your original equation. I'm again gonna use the factored form because I think it'll make it a little bit easier. So here I have 3 \times -3 \times (-3 + 2)^2 equals this 3 \times -3. I can go ahead and combine that. That's gonna give me -9. And then I have -3 + 2, which is going to give me -1, and that is squared. Now -9 times -1 squared. -1 squared is just 1, so this is just -9. And that's my first ordered pair, -3, -9. Now for my next ordered pair, I'm gonna calculate f(-1) by plugging -1 into my function. So 3 \times -1 \times (-1 + 2)^2. And simplifying that, 3 \times -1 is going to give me -3. And then -1 + 2 gives me +1, and that is squared. One squared is just a one, so this is just -3 times 1, which is -3. And here's my second ordered pair, -1 -3. Now lastly, I need to go ahead and calculate f(1). So plugging 1 into my function, I get 3 \times 1 \times (1 + 2)^2. And simplifying that, 3 \times 1 is just 3. 1 +2 is also 3, but that 3 is squared. So this becomes 3 \times 9 because 3 squared is 9. 3 \times 9 is 27. So we have our final ordered pair here, 1, 27. So we're done calculating those points. Finally, let's go ahead and put them up on our graph here. So my first point is -3, -9, which I can go ahead and plot -3, and then -9 is right around here. And then my second point I have is -1, -3, which plotting that -1 and -3. So right about in the middle there. And then finally 1, 27. So we see that this increases really quickly on this side, 1 all the way up to 27. Now filling in those unknown behavior I can now finally connect everything with a smooth and continuous curve like I know all polynomial functions have. So let's go ahead and connect everything using what we already know. So we know that our end behavior is gonna go down to negative infinity on this side, and then we're gonna go through this point and cross and all the way up. Now remember, it's okay if your curve isn't perfect. It can be really hard to draw these in a continuous line. I'm gonna go ahead and remove all of that extra stuff on my graph so I'm just left with my final graph. Remember the other step is just to help us later on so that we can really easily sketch this at the end. So here's my polynomial function. We still have one last thing to check here and that is our turning points, which we want to make sure does not exceed our maximum number. Now for our turning points, we wanna take our degree, which is 3 here, subtract 1, which gives us 2, and check that we don't have more than 2 turning points here. So I see that I have a turning point right at -2 and then somewhere around here is another turning point. I don't have any more here, so it looks like I'm good. I have not exceeded my maximum number of turning points. Now we have finished everything with our graph, but we're gonna take a look at our graph and determine 2 more things, our domain and our range. Now remember the domain, we actually don't have to determine anything because the domain of all polynomial functions is the same. It is negative infinity to infinity, or all real numbers. Now for our range, we do want to take a look at our graph here because it can vary based on our function, but we know that it continues on to negative infinity down here on the bottom. And then on the top, we see that this also carries on to positive infinity. So our range is actually going to be the same as our domain here and go from negative infinity to positive infinity. Now that's not always going to happen with every single polynomial function, it just happens to happen with this function. So we finally have graphed our full polynomial function and identified our domain and our range. Let's get some more practice.