in this video we're going to talk about how to graph polynomial functions using things such as end behavior multiplicity and even finding the zeros so let's begin by taking down some notes first let's begin our discussion with the graph of y equals positive x squared so this is an even function with a degree of two even functions are symmetrical about the y axis and so this is the shape of the y equals x squared graph the end behavior towards the left is up and the end behavior towards the right is up now there's other ways in which you can describe the end behavior of this graph on the right side as x approaches positive infinity y approaches positive infinity so here's how you can visualize it as you move towards the right along the x-axis as you go towards positive infinity the values along the y-axis are increasing you're going up towards positive infinity on the y-axis now the end behavior for this graph on the left as x approaches negative infinity y approaches positive infinity as we go to the left we can see the y values are increasing so this is another way in which you can describe the end behavior of that graph now let's consider another example in this case it's going to be y equals negative x squared now this is also an even function so we're going to have symmetry along the y-axis but we have a negative sign the leading coefficient is negative so this is going to reflect over the x-axis so the graph is going to look like this as we could see the end behavior on the left is down and it's also down on the right but we can express it this way just like before so on the left side as x approaches positive infinity that is as we go towards the right what happens to the y value well we could see that the y value is decreasing it's becoming more negative so it's approaching negative infinity so that's the end behavior i put left this should be on the right side because x is going towards positive infinity so that's the end behavior towards the right side now towards the left side that is when x goes to negative infinity so we're going towards the left notice what's happening to the y values they're also becoming more negative so y is approaching negative infinity so that's one way in which you could describe the end behavior of a particular graph now for the sake of practice go ahead and describe the end behavior for these two graphs y equals positive x cubed and y equals negative x cubed and i'll give you the shape of these graphs so positive x cubed looks like this it starts from the bottom and then let me draw this better it curves towards zero it's actually almost horizontal at the origin and then it goes back up so right here you have like a horizontal tangent now for y equals negative x squared it reflects about the origin so it looks like that and we still have that same horizontal tangent where it touches the x-axis so go ahead and describe the left and the right end behavior for these two graphs so let's start with the left side as x approaches negative infinity what happens to y well as we go towards the left we can see that the y values are becoming more negative y approaches negative infinity now for the right side as x goes to positive infinity y goes up it goes towards positive infinity as well so that's the end behavior for positive x cubed now let's do the same for negative x cubed as x approaches negative infinity what happens to y as we go towards the left the y values are increasing in the positive y direction so it goes towards positive infinity it's going up and for the right end behavior as x approaches positive infinity as we move towards the right side we can see that the curve is going down so y approaches negative infinity so now let's put everything together let's start with the even functions so y equals positive x squared what you need to know is that the end behavior is up up now the same is true for y is equal to positive x to the fourth this graph looks similar the end behavior is up on the left and up on the right so anytime you have a polynomial with an even degree and a positive leading coefficient the end behavior will be up up now if we have negative x squared that does not look like a 2 let's do that again the end behavior is going to be down on the left and down on the right so i'm just going to use d to represent down now for y equals negative x to the fourth the situation is similar it's going to be down on the left and down on the right as well so anytime you have an even exponent it's either up up or down down the difference is based on the sign of the leading coefficient if it's negative it's down down but if the lean coefficient is positive it's going to be up up now let's consider the examples where the exponent is odd so going back to positive x cubed the end behavior is down on the left up on the right now this is similar to y is equal to positive x to the first power but this is going to be a straight line but the end behavior is still the same it's down on the left up on the right so anytime the exponent is odd the end behavior is going to alternate it's going to be either down up or up down if the leading coefficient is positive it's going to go from down to up so it's going to be increasing now if the leading coefficient is negative we're going to get the reverse situation so instead of going from down up it's now going to be up and then it's going to go down and the same is true for y equals negative x to the first power it's a straight line but it's going to go in this direction it's initially up on the left but it's going to be down on the right so now let's summarize this in a table you may want to write this down so when the exponent or the overall degree of the polynomial function is even and if the leading coefficient is positive the end behavior will be up up if we have a degree that's even but the leading coefficient is negative then it's going to be down down that's the end behavior now if we have an odd degree with a positive leading coefficient it's going to be down at first and then it's going to be up if it's odd or negative it's going to be up and then down now let's talk about the behavior of the graph near an x-intercept i'm going to explain this more late in this video but for now you want to write this down so let's say the multiplicity of a particular x-intercept or particular zero is one this is going to look like the graph y equals x to the first power so at that zero it's simply just gonna cross the x axis basically in a straight line now the angle of the straight line can vary so you don't have to be too specific about that if the multiplicity is 2 it's going to look like the graph y equals x squared it's not gonna cross the x-axis but it's going to touch or bounce on the x-axis much in a similar way to the graph y equals x squared if the multiplicity is 3 for a particular 0 or x-intercept it's going to look similar to x cube but at at that x intercept so if you think of the graph of x cube it looks something like this so you're going to have that same shape here but you're going to have the behavior at that 0 is going to be horizontal so make sure you understand that if the multiplicity is 1 the behavior at the x intercept is going to be like a straight line it's going to cross the x axis in a straight line if the multiplicity is 2 at that x-intercept it's not going to cross the x-axis but it's going to bounce on you know that point and then if the multiplicity is 3 it's going to cross the x axis but not in a straight line but rather with a horizontal tangent so now let's work on some example problems consider the polynomial function y is equal to x plus 2 times x minus 1 squared times x minus 4. so we're given a polynomial function not in standard form but in factored form how can we graph this particular function well the first thing we need to do is we need to identify the zeros that is the x-intercepts so we're going to set y equal to 0 and we're going to solve for x so in this equation what are the possible solutions according to the zero product property in order to solve it we need to set each factor equal to zero if one of these factors is zero the entire expression on the right side will equal zero because zero times anything is zero so we're gonna set x plus two equal to zero we're gonna set x minus one equal to zero and x minus four for the first equation if we subtract two from both sides we'll get that x is equal to negative 2. for the second one if we add 1 to both sides we'll find that x is equal to positive 1. and here we just need to add 4 to both sides and then we'll get x is equal to four so that's how we could find the zeros or the x-intercepts of the polynomial function so all we need to do to quickly find the zeros is simply replace the sign if you see x plus two the zero is going to be negative two if you see x minus one the intercept is going to be positive one here x minus four just change negative four to positive four and that will give you the x-intercepts now the next thing we need is the multiplicity of each zero the multiplicity is going to be equal to the exponent of each factor so for this zero the multiplicity is going to be what we see here positive one for the factor x minus one or for the zero x equals one the multiplicity is going to be two and for the last one it's 1. and so now we have everything that we need in order to graph this polynomial function so let's go four units to the left and four units to the right so the first x-intercept on the left is that negative two the second one is at positive one and the last one is at four now there's one more thing that we need we need to determine the overall degree of the polynomial to find the overall degree simply add the exponents one plus two plus one is four so we have a degree four polynomial if you were to foil this entire expression the first term will be x to the fourth and it's positive all of the signs here in front of x are positive so we have a polynomial with an even degree and a positive leading coefficient what is the end behavior for this polynomial if it's even it's either up up or down down because this is positive the end behavior will be up up so let's put an up arrow above the first x-intercept and another up arrow above the last one so now we're going to sketch the graph from one arrow we're going to try to connect it to the other arrow so let's approach the first x-intercept now what is the behavior of the curve at this x-intercept in order to find the the behavior around that x-intercept we need to look at the multiplicity the multiplicity is one so the behavior is going to be similar to y equals x to the first power all we're going to do is draw a straight line through that x-intercept it's going to be linear at that point so because the multiplicity is 1 we're simply going to cross the x-axis now we need to turn towards the next x-intercept which is that positive one for this one the multiplicity is two so think of the graph y equals x squared the behavior of the curve at the x-intercept going to look like this so therefore we're going to bounce off the x-intercept so we're going to draw something that looks like a y equals x squared graph there but upside down and now for the last x-intercept the multiplicity is one so we're going to cross the x-axis with a straight line and so that's it that's how we can graph for this particular polynomial function now the only thing we don't know is how low the peak is or how high it is and the only way to get a good idea of the heights of those peaks is to plug in numbers so we're gonna try negative 1 and positive 2. and let's see what the y values are for those numbers so let's plug in negative 1 into the original expression so we're going to have negative 1 plus 2 and then negative one minus one squared and then negative one minus four negative one plus two is one negative one minus one is negative two and negative one minus four is negative five negative two squared is positive four four times negative five is negative twenty so that's the y value for that point now let's see what it is for two so this is going to be two plus two to the first power two minus one to the second power and then two minus four two plus two is four two minus one is one two i feel like i'm missing something here two minus negative four is negative two so for this one we get negative eight which is not as low as negative twenty let's try this one let's try one actually you know what i put the wrong number i take that back i see the mistake i made this is 3 and this intercept should be 4. so this should be 3 plus two and then three minus one squared and then three minus four three plus two is five three minus one is two three minus four is negative one two squared times four times negative one that's negative twenty okay so these two peaks are of the same height for this particular example and you can see why there's symmetry so to speak these two zeros have the same multiplicity and the distance between these two x-intercepts they're the same so due to the fact that this graph is symmetric about x equals 1 we can see why these two values are the same now this is not going to be the case for every other example it's just for this particular example so the best way to determine the relative heights of the peaks is simply to plug in numbers in this case they just happen to be the same