How do you graph #f(x)=(x^2-1)/x# using holes, vertical and horizontal asymptotes, x and y intercepts?

Question

Julia Adams · Accepted Answer

You have a vertical asymptote at x=0 because that would make the denominator equal zero. There's a slant asymptote at #y=x# because #x^2/x=x#

You know this graph can't exist at #x=0#, since that would make the denominator equal #0#. Because the polynomial on the top is of a bigger degree, there is a slant asymptote. Dividing the initial terms, we get #x^2/x=x#, so there is a slant asymptote at #y=x#. Since we can factor the top into #(x-1)(x+1)#, we know the function has two solutions at #x=+-1#.

Plotting these solutions and following the asymptotes makes this a straightforward graph to sketch: graph{(x^2-1)/x [-10, 10, -5, 5]}

Also, not all rational functions are so easy to predict the behavior of, so creating a table of x and y values is always a good idea! And if you need more information about how to find the asymptotes, look here.

Nathan Axel · Answer

undefined To graph the function f(x) = (x^2-1)/x, we can analyze its key features:

1. Holes: The function has a hole at x = 1, as the denominator becomes zero at this point. To find the y-coordinate of the hole, substitute x = 1 into the function: f(1) = (1^2-1)/1 = 0. Therefore, the hole is located at (1, 0).

2. Vertical Asymptotes: Vertical asymptotes occur when the denominator of the function becomes zero, but the numerator does not. In this case, the vertical asymptote is x = 0.

3. Horizontal Asymptotes: To determine the horizontal asymptote(s), we examine the behavior of the function as x approaches positive or negative infinity. Divide the leading terms of the numerator and denominator: (x^2/x) = x. As x approaches infinity, the function approaches positive infinity, and as x approaches negative infinity, the function approaches negative infinity. Hence, there are no horizontal asymptotes.

4. x-intercepts: To find the x-intercepts, set f(x) = 0 and solve for x: (x^2-1)/x = 0. This equation is satisfied when x = ±1. Therefore, the x-intercepts are (-1, 0) and (1, 0).

5. y-intercept: To find the y-intercept, substitute x = 0 into the function: f(0) = (0^2-1)/0 = undefined. Hence, there is no y-intercept.

By considering these features, you can plot the graph of f(x) = (x^2-1)/x accurately.