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

Video Transcript

Consider the graph of the function 𝑓 of 𝑥 is equal to one divided by 𝑥 plus two. What happens to the function when the value of 𝑥 approaches negative two? Option (A) the value of 𝑦 approaches ∞ when 𝑥 gets closer to negative two from the positive direction and approaches negative ∞ when 𝑥 gets closer to negative two from the negative direction. Option (B) the value of 𝑦 approaches ∞ when 𝑥 gets closer to negative two from the negative direction or from the positive direction. (C) The value of 𝑦 approaches negative ∞ when 𝑥 gets closer to negative two from the negative direction or from the positive direction. Or is it option (D) the value of 𝑦 approaches negative ∞ when 𝑥 gets closer to negative two from the positive direction and approaches ∞ when 𝑥 gets closer to negative two from the negative direction?

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

However, this is not necessary because we’re given a graph of the function. And even if we weren’t given a graph of the function, we could just sketch this graph by noting it’s a translation of the graph of one over 𝑥 two units to the left. And in the graph of a function, the 𝑥-coordinate of any point on the curve tells us the input value and the corresponding 𝑦-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 𝑥 approach negative two by seeing what happens to the 𝑦-coordinates of points on the curve as the input values of 𝑥 approach negative two from either direction.

To do this, let’s start by sketching the vertical line 𝑥 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. Let’s now see what happens to the outputs of the function as 𝑥 approaches negative two from either side. Let’s start with the positive direction. And remember this is the side from the positive values of 𝑥. As our values of 𝑥 approach negative two, we can see that the output values, that’s the 𝑦-coordinates of the points on the curve, are getting larger and larger. We can in fact see that the 𝑦-coordinates are growing without bound. So, as our values of 𝑥 get closer to negative two from the positive direction, the 𝑦-values are approaching ∞.

We can do the exact same thing to determine what happens to the outputs of the function as the values of 𝑥 get closer to negative two from the negative direction. This time, the 𝑦-coordinates of the points on the curve are decreasing without bound. So they’re approaching negative ∞.

And of the four given options, we can see that this only matches option (A). The value of 𝑦 approaches ∞ when 𝑥 gets closer to negative two from the positive direction and approaches negative ∞ when 𝑥 gets closer to negative two from the negative direction.