Determine the graphs end behavior find the x intercepts and...

determine the graphs end behavior. find the x intercepts and y intercept. determine whether the graph has symmetry. determine the graph of the function f(x)= x^3 +2x^2 - x -2 a. use the leading coefficient test to determine the graphs end behavior. which statement describes the behavior 1. the graph falls to the left and rises to the right 2. the graph falls to the left and to the right 3. the graph rises to the left and falls to the right 4. the graph rises to the left and to the right b. what are the x intercepts x= at which x intercepts does the graph cross the x axis? select the correct choice and if necessary fill the answer in the box to complete your choice 1. x = 2. there are no x intercepts which cross the x axis at which x intercepts does the graph touch the x axis and turn around. select the correct choice below and if necessary fill in the answer box to complete your choice 1. x = 2. there are no x intercepts at which the graph touches the x axis and turns around c. find the y intercept by setting x equal to 0 and computing f(0) d. determine whether the graph has y axis symmetry, origin symmetry, or neither. choose the correct answer below 1. origin symmetry 2. y axis symmetry 3. none of the above

Transcript

00:01 We were given the function f of x is equal to x cubed plus 2x squared minus x minus 2.

00:09 And we are asked to determine the graph's n behavior, x intercepts, y intercept, and whether it has any symmetry.

00:18 So let's get started.

00:21 Let's first talk about the graph's n behavior.

00:24 Now when you're thinking about n behavior, first of all, you take a look at the degree of the polynomial.

00:31 So this particular polynomial is a third degree.

00:35 So it's an odd degree polynomial, meaning that we have two options.

00:41 Either the end behavior looks something like this, where it goes up to the left, down to the right, and something happens in between.

00:52 Or it goes down to the left and up to the right, and something happens in between.

00:58 In other words, the two ends on the left hand and on the right, hand side, they are going to be opposites of each other.

01:06 That's what you get when you have an odd polynomial.

01:10 Now, which one of these happens is determined by the sign of the leading coefficient.

01:18 So looking at this, our leading coefficient here is actually one, positive one.

01:25 Now, since this is positive, this tells us that we have this situation right here, that we have the going down to the left and then going up toward the right.

01:43 Okay, so that takes care of the end behavior.

01:47 Now let's work on the intercepts.

01:50 I think i'll work on the x intercepts first.

01:53 Basically for the x intercepts, you need to solve the equation where the polynomial is equal to zero.

02:00 So we have zero is equal to x cubed plus 2x squared minus x minus 2.

02:07 Now luckily, i'm looking at this polynomial, and i think we can factor this.

02:13 We can first factor by doing a little bit of grouping here.

02:18 So i'm noticing that with the first two terms, i can maybe factor out in x squared, and i'll have x plus 2.

02:27 And from the second two terms, i can maybe factor out a negative 1.

02:34 So i'll have minus 1 times x plus 2 again.

02:41 This x plus 2 is showing up twice, which is really nice, meaning that i can go ahead and use that as an overall factor, x plus 2.

02:51 And i'll have x squared minus 1 left over.

02:56 And x squared minus 1 is a difference of two perfect squares.

03:00 So i can go ahead and go a little further with that.

03:03 I'll have x plus 1 and then x minus 1.

03:10 So from here, we can use the zero product property, which tells you.

03:15 Tells us that we can set each of these individual factors equal to zero.

03:25 So we'll have x intercepts at negative 2, negative 1, and positive 1.

03:33 Now the question is, what happens at these x intercepts? does the graph cross the x -axis, or does it touch and then go back around? well, this is where we can talk about multiplicity.

03:49 And we can just take a look at our factored form right here.

03:54 Multiplycity, we can tell the multiplicity of an x intercept by taking a look at if we have any exponents attached to these factors.

04:04 So as you notice, they're all x plus two to the first power.

04:08 I can maybe write that and it doesn't change anything.

04:11 X plus one to the first power.

04:13 Again, it doesn't change anything...