This problem is about finding the horizontal, slant, and vertical asymptotes, holes, domain in interval notation, x and y intercepts of rational functions, and graphing functions with all parts. The first step to this equation is noticing that the numerator has a greater degree than the denominator. This means that we go on to find the slant asymptote by using long division. We stop once you receive a y=mx+b equation. Next you find vertical asymptotes. This requires factoring and cancelling out any common factors from the polynomial. You set the remaining factors equal to 0 and solve. This number is your vertical asymptote value which is then used in limit notation. Next, you use the cancelled factors and set them equal to zero to find any holes in the graph. The domain comes fro the vertical asymptote value and is then used in interval notation to decide the direction of the graph. You find the x-intercepts by setting the simplified numerator equal to zero and solving for the x's. The y-intercepts are found by replacing the x's of the polynomial with zeroes.Then you graph the polynomial!
Some things that the problem solver needs to pay attention to are that they respond properly to the degrees! For example, in this problem, there was no horizontal asymptote because the degree of the numerator was larger than that of the denominator. Also, they have to factor correctly so that they can properly find the vertical asymptotes, domain, and holes! Lastly, they have to make sure that their graph does not touch the asymptotes of the graph.
Some things that the problem solver needs to pay attention to are that they respond properly to the degrees! For example, in this problem, there was no horizontal asymptote because the degree of the numerator was larger than that of the denominator. Also, they have to factor correctly so that they can properly find the vertical asymptotes, domain, and holes! Lastly, they have to make sure that their graph does not touch the asymptotes of the graph.
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