I am a bit confused about vertical asympotes. These two examples show different ways of getting asympotes. If I was to follow the first example to get the second example's asympote, it would not work out. Which example is correct, or are they both?
1. 2x/x^2-1 becomes factored to equal: 2x/(x-1)(x+1). So the asympotes are -1 for the first parenthesis and 1 for the second. The asympote was found by taking the number beside the x.
2. x/x^2-2x-3 factors to x/(x-3)(x+1) The asympotes according to the algebra book are three and -1. They were found not by taking the number beside the x as in the previous example, but by adding the number that would equal zero in the place of the x.
Which method is correct?
Thanks for your help -Gina
Those actually both use the same method, the one you described for the second example. x=(-1) makes (x+1) equal to zero, and x=1 makes (x-1) equal to zero.
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Both methods you mentioned can be used to find the vertical asymptotes of a given function, depending on the form of the rational function. Let's break it down:
1. Method 1: Finding vertical asymptotes by equating denominators to zero.
In this method, you factor the denominator and set each factor equal to zero to find the vertical asymptotes. This method works well when the denominator is in factored form.
Example: 2x/(x-1)(x+1)
Here, the factors of the denominator are (x-1) and (x+1). Setting them equal to zero gives you x-1=0 and x+1=0. Solving for x, you find that the vertical asymptotes are x=1 and x=-1.
2. Method 2: Finding vertical asymptotes by identifying the values that make the denominator zero.
In this method, you identify the values of x that would make the denominator zero. This method works when the denominator is not in factored form.
Example: x/(x-3)(x+1)
Here, you observe that the denominator is not factored. However, by setting the denominator (x-3)(x+1) equal to zero, you can find the values that make it zero. Solving (x-3)(x+1)=0, you find that x=3 and x=-1.
Both methods are correct and can be used depending on the form of the rational function. Method 1 is more suitable when the denominator is factored, while Method 2 can be used when the denominator is not factored or given in factored form. It is essential to adapt your approach accordingly.