Work each problem. Choices A–D below show the four ways in which ...

Question

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d).ƒ(x)=-1/(x-2)

Accepted Answer

Hey, everyone in this problem, we're asked to identify the graph of the following rational function through the way it approaches the vertical Asymptote X equals four. The function given is F of X is equal to negative one divided by X minus four. We're given four answer choices. They're all graphs and they all show a vertical Asymptote at X equals four. Option A has the graph increasing to positive infinity on both the left and right sides of the vertical Asymptote. Option B has the graph decreasing to negative infinity on both the left and right side of the vertical ascent to option C has the graph increasing to positive infinity as it approaches the vertical Asymptote at X equals four on the left hand side and on the right hand side, decreasing to negative infinity. And option D as a graph approaching negative infinity from the left hand side of the vertical Asymptote and positive infinity. From the right hand side. It's what we wanna do here is we wanna look at our function and we want to think about what is happening to the value of this function on either side of that vertical Asymptote. OK. So we approach the vertical as to from the left hand side, what happens? Well, if we're looking at X values that are to the left of X equals four, those X values are going to be less than four. So for X value is less than four, then X minus four will be negative. And so our function F of X will be equal to a negative value divided by another negative value and which is going to give us a positive value. So to the left of our X A or our Asymptote, sorry, our vertical Asymptote at X equals four as we get closer and closer and closer to X equals four. The denominator is gonna get closer and closer to zero. And it's going to impro approach either positive or negative infinity. And when we look at it from the left, because the X values are going to be to the left of X equals four. OK? They're gonna be less than four, then X minus four is going to be negative. We get a negative value divided by a negative value in our function. And so the function is going to be positive. OK. So we're gonna be approaching positive infinity on the left hand side. And if we approach from the right, we're gonna look at this the same way. So we're approaching from the right, the X value is to the right of X equals four. That means it's bigger than four. So X minus four is going to be positive since X is bigger than four, when we take our function F of X, we get a negative value. OK? Negative one in the numerator divided by a positive value in the denominator which is going to give us a negative value. And so the denominator is still getting really, really close to zero. So we're going off to positive or negative infinity in this case, because X is bigger than four, X minus four is gonna be positive. And so our function ends up being negative. And so in this case, we're going off to negative infinity as we approach from the right hand side. So if we compare this to our answer traces, we found that from the left hand side of our Asymptote, we're approaching positive infinity from the right hand side of our Asymptote, we're approaching negative infinity. And so this corresponds with option C. On the left hand side of this Asymptote, this function is positive and increasing. And on the right hand side, as we approach that vertical Asymptote, the function is negative and decreasing. Thanks everyone for watching. I hope this video helped see you in the next one.

Work each problem. Choices A–D below show the four ways in which the graph of a rational function can approach the vertical line x=2 as an asymptote. Identify the graph of each rational function defined in parts (a) – (d).ƒ(x)=-1/(x-2)

Key Concepts

Rational Functions

Vertical Asymptotes

Graph Behavior Near Asymptotes

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