Graph Asymptotes and Intercepts of 1/(x-2) and 1/x - 2

In this article, we will explore the asymptotes and intercepts of the functions y 1/(x-2) and y 1/x - 2. Understanding these aspects of graphing functions is crucial for advanced mathematics, particularly in calculus and algebra.

Understanding Asymptotes and Intercepts

Asymptotes are lines that a curve approaches but never touches, while intercepts provide points where the graph intersects the axes. We will focus on vertical and horizontal asymptotes, and both the x-intercept and y-intercept for the given functions.

Case 1: y 1/(x - 2)

This function has a vertical asymptote at x 2 because the denominator becomes zero, making the function undefined. There are no horizontal asymptotes in this case. To find the y-intercept, we set x 0:

[ f(0) frac{1}{0-2} -frac{1}{2} ]

For the x-intercept, we set y 0 and solve for x:

[ 0 frac{1}{x-2} ]

No real solution for this equation, indicating no x-intercept.

Case 2: y 1/x - 2

This function can be interpreted in two ways:

y 1/x - 2 y (1/x) - 2

We will consider both interpretations.

Interpretation 1: y 1/x - 2

The y-intercept occurs when x 0:

[ y frac{1}{0} - 2 ]

This is undefined, so there is no y-intercept. For the x-intercept, set y 0:

[ 0 frac{1}{x} - 2 ] [ frac{1}{x} 2 ] [ x frac{1}{2} ]

The x-intercept is at x 0.5.

Interpretation 2: y (1/x) - 2

This function has a vertical asymptote at x 0 because the denominator becomes zero. The horizontal asymptote is at y -2:

[ lim_{x to pminfty} y lim_{x to pminfty} frac{1}{x} - 2 -2 ]

For the y-intercept, we set x 0:

[ y frac{1}{0} - 2 ]

Again, this is undefined, so there is no y-intercept. For the x-intercept, set y 0:

[ 0 frac{1}{x} - 2 ] [ frac{1}{x} 2 ] [ x frac{1}{2} ]

The x-intercept is at x 0.5.

In summary:

y 1/(x-2) has a vertical asymptote at x 2 and no horizontal asymptote. No intercept. y 1/x - 2 (interpreted as (1/x) - 2) has a vertical asymptote at x 0 and a horizontal asymptote at y -2. No intercept.

Graphing the Functions

To visualize these functions, we can use Desmos or any other graphing tool. Here are the graphs of the two functions:

y 1/(x-2) - Graph y 1/x - 2 - Graph

For more detailed analysis and further reading, you can refer to advanced mathematics textbooks or online resources dedicated to calculus and graphing functions.