Graph the polynomial function. Determine the domain and range. f ( x ) = ( 3 x + 2 ) ( x − 1 ) 2 f\left(x\right)=\left(3x+2\right)\left(x-1\right)^2

Hey everyone, let's graph this polynomial function. Here I have the function f of x is equal to 3x plus 2 times x minus 1 squared. Let's jump right into finding the end behavior of our polynomial. Now to do this we need to look at our leading term and since my function is actually already factored I could fully expand this to find my leading term in standard form or here I'm going to show you a shortcut to more quickly find your leading term without having to fully expand. So I'm gonna take my x term of each factor and however many times that factor occurs, this one only occurs once. So this is 3 x, and then I have this x which is squared. So 3x times x squared is going to give me 3x cubed. So this shows me what my leading coefficient would be if I were to fully expand this out without having to do that. So with this, now I can look at my leading coefficient 3, which is positive, which tells me my right side is going to rise. Then looking at my degree here, which is 3, an odd number, that tells me that my n's are going to have the opposite behavior, odd, opposite. And now I can go ahead and plot this on my graph or sketch this. So my right side is going to rise, then my left side is going to have the opposite behavior. So that's my end behavior. Let's move on to find our x intercepts. Now to find our x intercepts, we wanna go ahead and solve the equation f of x is equal to 0. And here my function is 3x+2 timesx minus 1 squared is equal to 0. So let's come down here and solve that equation. Now since this is already factored, I can go ahead and take each factor and set them equal to 0. So 3x +2equals0, and then x minus 1 equals 0. So if I subtract 2 from each side, I'm left with 3x is equal to negative 2. Then dividing both sides by 3, that cancels, leaving me with x is equal to negative 2 third, and that is my first x intercept. Now adding 1 to both sides on this factor, I'm left with x equals 1, and that is my other x x intercept. So I have x is equal to negative 2 thirds and x is equal to 1. So now that I have my x intercepts, I want to also determine their multiplicity so that I can know whether they they touch or cross the x axis at those points. So we wanna look at the number of times each factor occurs. So with my factor x equals negative 2 thirds, this comes from my factor 3x plus 2, which only happens once, so it has a multiplicity of 1. Now 1 is an odd number, which tells me that my graph is going to fully cross the x axis at this point. Then looking at my other factor x is equal to 1, this comes from x minus 1 which is squared, so it has a multiplicity of 2, an even number. So my graph is simply going to touch the x axis at this point. So let's go ahead and sketch all of this on our graph. So I have negative 2 thirds, which is right about here, has a multiplicity of 1, which means we're gonna cross the x axis at that point. And then x equals 1 where we are simply going to attach the x axis x axis at that point. Let's go ahead and look at our our y intercept now. So we wanna compute y or f of 0 by plugging 0 into our function. So since my function is already factored, I'm gonna just use the factor form. 3 times 0, plugging 0 in for x, plus 2 times 0 minus 1 squared. Now 3 times 0 is just a 0, so this just leaves me with a 2. And then 0 minus 1 gives me negative one, and that is squared. Now 2 times negative one squared is just 2 times 1, and 2 times 1 is just 2. So that is my y intercept, y equals 2, which I can go ahead and plot on my graph as well at y equals 2. Okay. Now that we have all of these going on, let's go ahead and determine our intervals and then plot a point in each of them. So going back up to my graph here, let's take a look at what our interval should be. Now looking at my graph, I go from left to right, so negative infinity until I reach my first known point, which is at this negative 2 thirds. So my first interval is from negative infinity to negative 2 thirds. Then from negative 2 thirds, I reach my next known point, which is at 0, so negative 2 thirds, until I reach 0. And then from 0, I reach 1, my next known point, so 0 to 1. 1. And then finally, 1 and beyond because I don't know anything else after 1. So those are my 4 intervals here. I'm gonna stop for a second because I wanna consider what's going on with these intervals. Now these two middle intervals, so from negative 2 thirds to 0 and from 0 to 1, these are really small intervals. So we need to think about if calculating a point here is really going to be worth it and is going to add a lot value to our graph here? And here, I would say no because these are so small and I already know the behavior of my zeros, whether it touches or crosses, it's not really gonna do a whole lot to calculate some small really small points within there. So we're gonna go ahead and skip that, and we're only going to calculate points for our outside intervals to give us more information there. Now, of course, if you are explicitly asked to calculate points in every interval, you should go ahead and do that. But here we're going to skip it because we already have a good idea of what our graph is going to look like there. So let's go ahead and calculate points within each of these outside intervals. So my first interval from negative infinity to negative 2 thirds, I'm going to choose an x value in that interval. And I'm going to go ahead and just choose negative one since it's right there. And then from 1 to infinity we want to choose another point. I'm simply going to choose 2 just so I know how quickly it's going to increase from there. So let's go ahead and plug this into our function in order to get those points. So f of negative 1, I'm gonna plug negative 1 into my function. So 3 times negative 1 plus 2 times negative one minus 1 squared. Now simplifying that, 3 times negative one is negative three plus 2. Negative one minus 1 is negative 2, so that's negative 2 squared. Then negative 3 +2 is going to give me negative 1. And then negative 2 squared is positive 4, so this is negative one times 4. I'm left with negative 4 for that ordered pair, negative one and negative 4. Then for my other point, I am looking at f of 2, plugging 2 into my function. So this is 3 times 2 plus 2 times 2 minus 1 squared. Now 3 times 2 is 6, so this is 6 +2. And then 2 minus 1 gives us 1 and that is squared. So I am left with 8 times 1 squared, which is just 1. 8 times 1 is just 8. So this ordered pair is 28. Now let's go ahead and plot that on our graph here. So negative one negative four plotting that. Negative one and negative four down here. And then 28. Now 8 is gonna be off my graph, which does give me more information. It tells me that my graph is gonna increase a lot here. So it still gives me a great idea of what my graph is going to look like. So now that we have all of this, we can go ahead and connect everything with a smooth curve. So I know we go down to negative infinity on this left side, and then we go up to our first x intercept at negative 2 thirds where we fully cross the x axis up to our y intercept and then back down to our next intercept where we simply bounce and touch the x axis, and then we increase rapidly because we know that that point at 2 is all the way off our graph, so we need to pull it all the way up here. Now don't worry if you can't get these lines perfectly straight or smooth, it can be really hard to draw some of these polynomial graphs, but I'm sure you're doing a great job. So we wanna check one more thing here, so I'm gonna erase those little sketches from the beginning, and then we're gonna go ahead and check for our turning points. Now our turning points are simply our degree minus 1. Our degree here is 3 because we found that leading term. 3 minus 1 gives me 2. Now looking at my graph, do I have more than 2 turning points? Well, I change direction here and here, and that's just 2. I don't change direction anywhere else so it looks like I'm good. I do not violate my maximum turning points. Now we want to check one more thing here, our domain and our range. Our domain, we don't even need to look at our graph. We know what it's going to be negative infinity to infinity. And then for our range, we want to take a look at our graph here. Now since it goes all the way to negative infinity over here and to positive infinity on the other side, we know that our range is going to be the same negative infinity to positive infinity, but you always want to double check depending on your function. So that's all for this one. Thanks for watching.