# Question Video: Describing the Asymptote of a Function from Its Graph Mathematics • 10th Grade

Consider the graph of the function π(π₯) = 1/(π₯ + 2). What happens to the function when the value of π₯ approaches β2?

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### Video Transcript

Consider the graph of the function π of π₯ is equal to one divided by π₯ plus two. What happens to the function when the value of π₯ approaches negative two? Option (A) the value of π¦ approaches β when π₯ gets closer to negative two from the positive direction and approaches negative β when π₯ gets closer to negative two from the negative direction. Option (B) the value of π¦ approaches β when π₯ gets closer to negative two from the negative direction or from the positive direction. (C) The value of π¦ approaches negative β when π₯ gets closer to negative two from the negative direction or from the positive direction. Or is it option (D) the value of π¦ approaches negative β when π₯ gets closer to negative two from the positive direction and approaches β when π₯ gets closer to negative two from the negative direction?

In this question, weβre given the graph of a function π of π₯ is equal to one divided by π₯ plus two. And since this function is the quotient of two polynomials, we can say π of π₯ is a rational function. We want to determine what happens to the graph of our function as our values of π₯ approach negative two. And thereβs a few different ways we could go about this. For example, we could determine what happens to the output of our functions around π₯ is equal to negative two directly by using the given function. For example, we could construct a function table with our values of π₯ getting closer and closer to negative two.

However, this is not necessary because weβre given a graph of the function. And even if we werenβt given a graph of the function, we could just sketch this graph by noting itβs a translation of the graph of one over π₯ two units to the left. And in the graph of a function, the π₯-coordinate of any point on the curve tells us the input value and the corresponding π¦-coordinate of this point tells us the output value of the function. So we can determine what happens to the output of this function as the values of π₯ approach negative two by seeing what happens to the π¦-coordinates of points on the curve as the input values of π₯ approach negative two from either direction.

To do this, letβs start by sketching the vertical line π₯ is equal to negative two onto the given diagram. We can see that the curve approaches this line. So this is a vertical asymptote of the function. Letβs now see what happens to the outputs of the function as π₯ approaches negative two from either side. Letβs start with the positive direction. And remember this is the side from the positive values of π₯. As our values of π₯ approach negative two, we can see that the output values, thatβs the π¦-coordinates of the points on the curve, are getting larger and larger. We can in fact see that the π¦-coordinates are growing without bound. So, as our values of π₯ get closer to negative two from the positive direction, the π¦-values are approaching β.

We can do the exact same thing to determine what happens to the outputs of the function as the values of π₯ get closer to negative two from the negative direction. This time, the π¦-coordinates of the points on the curve are decreasing without bound. So theyβre approaching negative β.

And of the four given options, we can see that this only matches option (A). The value of π¦ approaches β when π₯ gets closer to negative two from the positive direction and approaches negative β when π₯ gets closer to negative two from the negative direction.