Sketching the Function f(x) (x - 1) / (x^2 - x - 6)

When you need to sketch the function f(x) (x - 1) / (x^2 - x - 6), it's important to carefully analyze its key features. This function has both vertical and horizontal asymptotes, and understanding these will help you create an accurate plot. Let’s go through the process step-by-step.

Domain Restrictions

First, we need to find any domain restrictions by setting the denominator equal to zero and solving:

x^2 - x - 6 0

Solving this, we get:

(x - 3)(x 2) 0

x 3, -2

These values create vertical asymptotes at x -2 and x 3. Therefore, the domain of the function is all x except x ≠ -2, 3.

Asymptotes

To determine the horizontal or slant/oblique asymptotes, we compare the degrees of the numerator and the denominator.

The degree of the numerator (1) is less than the degree of the denominator (2), so the horizontal asymptote is y 0.

Next, let's sketch out the function by plotting points around the asymptotes to get a more precise understanding.

Plotting Points and Asymptotes

Let's break it down into intervals and sketch the curve:

Interval: x

Choose an x value slightly less than -2, say x -2.1.

f(x) (x - 1) / (x^2 - x - 6)

f(-2.1) (-2.1 - 1) / ((-2.1)^2 - (-2.1) - 6) -3.1 / (4.41 2.1 - 6) -3.1 / 0.51 ≈ -6.08

The function is negative and the curve goes down to negative infinity as it approaches the vertical asymptote at x -2.

Interval: -2

Choose an x value between -2 and 1, say x 0.

f(0) (0 - 1) / (0^2 - 0 - 6) -1 / -6 1/6 ≈ 0.167

The function is positive, and the curve moves from negative infinity as it approaches x -2 to the point (0, 1/6).

Interval: 1

Choose an x value between 1 and 3, say x 2.

f(2) (2 - 1) / (2^2 - 2 - 6) 1 / (4 - 2 - 6) 1 / -4 -0.25

The function is negative, and the curve moves from the point (2, -0.25) to negative infinity as it approaches the vertical asymptote at x 3.

Interval: x > 3

Choose an x value greater than 3, say x 4.

f(4) (4 - 1) / (4^2 - 4 - 6) 3 / (16 - 4 - 6) 3 / 6 0.5

The function is positive and the curve moves from negative infinity as it approaches x 3 to the point (4, 0.5).

Graphing the Function

To graph the function, you can use Excel or any graphing software. Set up a table with values of x from -2.1 to 4 in small increments (e.g., 0.1). Then calculate the corresponding y values using the function formula.

The graph will show vertical asymptotes at x -2 and x 3, and a horizontal asymptote at y 0. The curve will approach these asymptotes as described above, with the appropriate sign changes.

By plotting these points and connecting them smoothly, you can create an accurate graph of f(x) (x - 1) / (x^2 - x - 6).