Question

Inverse Variation GraphsInverse variation equations are special types of rational equations. Rational equations are equations that have polynomial denominators. In other words, these types of equations have variables in the denominator. One of the features of rational equation graphs is asymptotes.An asymptote is a line that defines a graph’s end behavior and values that the graph will not cross. There are two types of asymptotes that will apply to inverse variation graphs: horizontal and vertical.Answer the following questions to graph the inverse variation equation y=1/x.Part ATo begin graphing the equation, fill in the table. X y-1000-100-10-101101001000Part BBased on the table you filled out, how do you expect the value of y to behave as x increases toward infinity and decreases toward negative infinity?Part CBased on your answer from part B, what is the equation of the horizontal asymptote of y=1/xPart DNext, let’s focus on how the graph behaves around the undefined value. Begin by filling in the following table. X y -1-0.1-0.01-0.00100.0010.010.11Part EBased on the table you filled out, as x gets closer to zero from the left and the right, how do you expect the value of y to behave?Part FBased on your answer from part E, what is the equation of the vertical asymptote of y=1/x?Part G Draw the graph of the equation by first graphing the asymptotes and then the graph of the equation. Part HYou’ve already learned that inverse variation equations will always be in the form y=k/x. Based on this, what can you say about the asymptotes for any inverse variation equation? Part IBased on the value of the constant of proportionality, k, explain how the graph of an inverse variation equation will look.

Answer 1

Part A

The table would be the following:

Part B

Notices that as x increases toward infinity and decreases toward negative infinity, y gets closer and closer to zero. Thereby, y tends towards 0 when x tends towards infinity and decreases toward negative infinity.

Part C

The horizontal asymptote would be:

`y=0`

Part D

The table would be:

Part E

As x gets closer to zero from the left, y tends towards negative infinity. As x gets closer to zero from the right, y tends towards infinity

Part F

The vertical asymptote is:

`x=0`

Part G

The graph of the function and the asymptotes is:

Part H

We can conclude that any inverse variation equations will have y = 0 as an horizontal asymptote and x = 0 as a vertical asymptote.

Part I

As the value of k increases, the graph will get wider an wider. For example, this is the graph for k = 10