How to Sketch a Graph: Intercepts, Asymptotes, and Symmetry for f(x) 1/(x-3)

Graph sketching is a fundamental skill for any mathematics student, particularly in the realm of calculus and analysis. The function f(x) 1/(x-3) is a hyperbola, which can be a bit more challenging to plot due to its vertical and horizontal asymptotes, intercepts, and symmetry. This guide will walk you through the process of sketching this graph, ensuring you understand each key concept involved.

Understanding the Function

The function f(x) 1/(x-3) has an interesting form that requires careful analysis. Let's break it down:

1. Vertical Asymptote

The vertical asymptote is where the function is undefined. In this case, the denominator (x-3) becomes zero, making the function undefined at x 3. Therefore:

Vertical Asymptote: x 3

2. Horizontal Asymptote

The horizontal asymptote indicates the behavior of the function as x approaches infinity. To find the horizontal asymptote, examine the degrees of the numerator and the denominator:

deg(numerator) 0, deg(denominator) 1

Since the degree of the numerator is less than the degree of the denominator, the function approaches y 0 as x → ∞ or -∞. Thus:

Horizontal Asymptote: y 0

3. X-Intercept (or Zeroes)

To find the x-intercept, set f(x) 0:

1/(x-3) 0

There is no x that can satisfy this equation because the fraction can never be zero. Therefore, there is no x-intercept.

No X-Intercept: None

4. Y-Intercept

To find the y-intercept, set x 0:

f(0) 1/(0-3) -1/3

Y-Intercept: (0, -1/3)

5. Symmetry

For some functions, symmetry can help simplify the sketching process. A point symmetry at (3, 0) means if you imagine a point (a, b) on the graph, then the point (6-a, -b) should also be on the graph. This suggests a rotational symmetry about the point (3, 0).

To confirm, consider the function f(x) and its reflection through the point (3, 0):

Let f(x) 1/(x-3). Reflect this through the point (3, 0):

For any point (x, y) on f(x), the reflected point would be (6-x, -y). We can verify this by substituting x-3 with 6-x-3, thus showing f(6-x) -y.

Center of Symmetry: (3, 0)

Sketching the Graph

Now that we understand the key features, we can sketch the graph:

Plot the Vertical Asymptote: Draw a dashed vertical line at x 3. Plot the Horizontal Asymptote: Draw a dashed horizontal line at y 0. Mark the Y-Intercept: Plot the point (0, -1/3). Use Symmetry: Knowing the center of symmetry at (3, 0), plot points symmetrically. For example, if (1, 1/2) is a point, (5, -1/2) should also be a point. Plot Additional Points: Choose values for x to plot more points and close the curve around the asymptotes. For instance, calculate f(4) 1, f(2) -1, and f(1) -1/2.

Finally, draw a smooth curve through these points, ensuring it follows the asymptotes' directions without crossing the asymptotes.

Conclusion

By understanding and identifying the vertical and horizontal asymptotes, intercepts, and symmetry, you can effectively sketch the graph of f(x) 1/(x-3). Remember that the process involves visualizing and plotting the key elements in a systematic manner, and the use of symmetry can simplify the task. Practice with different functions will refine your skills, making graph sketching a valuable tool in your mathematical arsenal.