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In this video we're going to talk about how to graph polynomial functions we're going to talk about how to determine the multiplicity the end behavior and how to find the zeros as well so let's go over some basics so let's say if the leading term is positive and if the exponent is even this graph has the general shape that looks like this it touches the x-axis and then it bounces off at the zero now the end behavior if you look at the arrows it's up on the left side and the arrow points up on the right side now as X approaches positive infinity notice what happens to the Y value so as X approaches infinity you travel on the right on the x axis notice that the curve it travels upward towards positive infinity so Y approaches positive infinity now what about the left side as X approaches negative infinity or if we travel towards the left of the graph what happens to the Y values Y also approaches positive infinity as you travel to the left the curve travels in the upward direction now what if the leading term is negative but the exponent is still even what type of shape will we have because the multiplicity which is the exponent since it's even just like the last example the curve is going to touch to zero but it's going to bounce it's not going to cross the x-axis so as X approaches positive infinity what happens to the Y value so as you travel towards the right on the x-axis notice that the curve travels in the downward direction that means y approaches negative infinity so notice that both arrows are facing down if it's even and it has a negative exponent the end behavior is down in Quadrant three and down in Quadrant four now what about as X approaches negative infinity as you travel towards the left along the x axis the Y value decreases so Y approaches negative infinity so that's the end behavior if the leading coefficient is negative and if the multiplicity is even or if the exponent is even now what about if we have a positive coefficient with an odd multiplicity how would you describe the end behavior in this particular example so for this graph it looks something like this it crosses the x axis since we have an odd multiplicity but for X cubed notice that the graph is it appears horizontal at the zero and then it increases so in this region at the zero its horizontal but the end behavior is down up so it's down in Quadrant three up in quadrant one now as X approaches positive infinity what happens - why does it approach positive infinity or negative infinity what would you say so as you travel to the right on the x axis notice that the Y value increases so goes up towards positive infinity and what about as X approaches negative infinity what value does Y approach so as you travel to the left notice that the curve decreases towards the negative y axis so approaches negative infinity so remember if it's positive and odd then the end behavior is down up now what if it's negative and odd so the end behavior is going to be up down so it's up towards quadrant two down towards quadrant four so as X approaches positive infinity notice that Y approaches negative infinity as you travel to the right the curve decreases towards the negative y axis and as X approaches negative infinity Y approaches positive infinity so as you travel towards the left on the negative axis the negative x axis the Y value goes up in the positive Y axis direction so it's a positive infinity so let's review what are the end behaviors for positive x to the first power and negative x to the first power so here on the left side the coefficient is positive but the exponent is odd so for this graph it's going to be down up the graph y equals x is a straight line it looks like that but notice that the arrow is down in Quadrant three and up in Quadrant four now for negative x to the first power it's like it's going to look like negative x cubed in terms of end behavior so just like negative x cubed it's up in quadrant two but down in project four so it's described us up down if you want now what if it's positive and even let's say X to the fourth or negative and even the end behavior for this if it's positive and even it's going to be up in Quadrant two and up in quadrant one if it's both negative it's going to be down down in Quadrant three and four so with this information we can now sketch a few polynomial functions so let's say if you get a function it looks like this y equals x plus two X minus one squared and let's say X line is four so the first thing you want to do is you want to set the function equal to zero and find the zeros find the x-intercepts because it's in factored form you can set each factor equal to zero so if X plus two is zero that means X is equal to negative two that's the first x-intercept if X minus one is equal to zero then X is equal to positive one that's the second x-intercept and if X minus four is equal to zero if you add four to both sides X is equal to positive four so those are the three x-intercepts that we have for this particular function so I'm just going to write it here negative two one and four so now let's sketch a rough graph the first thing you should do is plot the x-intercepts so we

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