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

Consider the graph of the function π(π₯) = 1/(1 β π₯) + 2. What is the end behavior of the graph as π₯ approaches 1? [A] The value of π¦ approaches β when π₯ gets closer to 1 from the positive direction and approaches ββ when π₯ gets closer to 1 from the negative direction. [B] The value of π¦ approaches ββ when π₯ gets closer to 1 from the positive direction and approaches β when π₯ gets closer to 1 from the negative direction. [C] The value of π¦ approaches β when π₯ gets closer to 1 from the negative direction or from the positive direction. [D] The value of π¦ approaches ββ when π₯ gets closer to 1 from the negative direction or the positive direction.

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### Video Transcript

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

In this question, weβre given the graph of a function: π of π₯ is equal to one divided by one minus π₯ plus two. And we need to use this graph or the given function to determine the end behavior of this graph as the values of π₯ approach one. And thereβs many different ways we could go about answering this question. For example, we can determine the end behavior of this graph as π₯ approaches one entirely by using the given function.

To see how we might do this, letβs consider the denominator in the given function: one minus π₯. If our values of π₯ are greater than one, which means π₯ is in the positive direction of one, then we can note one minus π₯ will be negative, since weβre subtracting a number bigger than one from one. We can also note as the values of π₯ get closer and closer to one, the magnitude of one minus π₯ will get closer and closer to zero. In other words, the distance between one and π₯ gets smaller as π₯ gets closer and closer to one. So weβre dividing one by a negative number with a very small magnitude. And as the magnitude of this number gets smaller, its reciprocal will get larger in magnitude. And adding two to this value wonβt change this fact. So π of π₯ will approach negative β as our values of π₯ approach one from positive direction.

However, this level of analysis is very difficult. So, instead, itβs very useful to be able to do this from a given diagram or, if possible, to sketch a diagram to help us determine this information. We can do this by recalling the input values of the function are the π₯-coordinates of the point on the curve and the output values are the corresponding π¦-coordinates. This means we can determine what happens to the outputs of the function as our values of π₯ approach one from the positive direction by seeing what happens to the π¦-coordinates of points which lie on the curve.

To do this, letβs add in the vertical line π₯ is equal to one onto our diagram. We can see that the curve gets closer and closer to this line. However, it never touches the line. This is a vertical asymptote of the curve. In particular, we can see as the values of π₯ approach one from the right, the π¦-coordinates of the points on the curve are going lower and lower. Theyβre unbounded, so theyβre approaching negative β. And we get a similar story if we look at the values of π₯ approaching one from the negative direction. We can see that the π¦-coordinates of the point on the curve are getting larger and larger, and theyβre unbounded. So theyβre approaching positive β.

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