Use analytical methods and/or a graphing utility to identify t...

Question

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.

f(x)=x^2−3x+2 / x^10−x^9

Accepted Answer

Welcome back, everyone. In this problem, we want to determine the vertical asymptotes of the function G of Y equals Y cubed minus 6Y + 4, all divided by Y 5 minus 2 Y 4th. A says Y equals 2, B Y equals 0, C Y equals 0, and Y equals 2, and D Y equals 3. Now, are we going to figure out where our vertical asymptotes are for this function? We'll recall that vertical asymptotes occur when a factor of the denominator of the expression does not cancel with a factor from the numerator. So essentially we'll need to figure out what values of Y make our denominator 0, and then see if that value also makes our numerator 0. If it makes the numerator 0, that means there is a hole, and if it does not, then it's a vertical asymptode. So let's set the denominator to 0. And set the denominator to 0, which means that we get y 5th minus 24th to be equal to 0. We can factor Y 4th from both terms, and we are left with Y 4 multiplied by Y minus 2 equals 0. And know what this means is that either Y 4th is equal to 0 or Y minus 2 equals 0. When Y 4 equals 0, if we find the 4th root of both sides, that makes Y equals 0. And if Y minus 2 equals 0, then Y equals 2. So these are the two values that will make our denominator 0. Now, how do they affect the numerator? Well, for our numerator. Which is Y cubed minus 6 Y + 4, OK. Then when Y is equal to 0, or numerator is 0Q minus 6 multiplied by 0 + 4. That's going to be 0 minus 0 + 4, which equals 4. Since this value is not equal to 0, then a vertical asymptode exists at Y equals 0. OK. So here is where one of our vertical asymptotes are. Next, let's test it when Y equals 2. No, our numerator is going to be 2 cubed minus 6 multiplied by 2 + 42 cubed is 86 multiplied by 2 is 12. 8 minus 12 is -4 + 4. Uh, which is going to give us 0, OK? Now, since our numerator is 0, that means a factor can be canceled from the numerator when Y equals 2, and thus there is a hole. So here's where there's a hole and not a vertical asymptote. Therefore, this tells us that Y equals 0 is the only vertical asymptote of our function. B is the correct answer. Thanks for watching everyone. I hope this video helped.

Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.
f(x)=x^2−3x+2 / x^10−x^9

Key Concepts

Vertical Asymptotes

Rational Functions

Factoring Polynomials

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