Question

Sketch the graph of the equation. Use intercepts, extrema, and asymptotes as sketching aids. \(f(x)=\frac{1}{x^{2}-x-2}\)

Short answer

The function \(f(x)= \frac{1}{x^{2}-x-2}\) has vertical asymptotes at \(x=2\) and \(x=-1\), a horizontal asymptote at \(y=0\), and a y-intercept at (0, -0.5). The function can be sketched using this information, with the curve approaching the asymptotes but never crossing them.

Step-by-step solution

Factor the Denominator

Firstly, the denominator of the function, \(x^{2}-x-2\), needs to be factored. The factors of the quadratic \(x^{2}-x-2\) are \((x-2)(x+1)\). So, \(f(x)= \frac{1}{(x-2)(x+1)}\).

Find the Vertical Asymptotes

Vertical asymptotes occur at the values of x that make the denominator zero. So, we need to find for which x-values \((x-2)(x+1)\) equals zero. These values are \(x=2\) and \(x=-1\). These x-values are the vertical asymptotes.

Find the Horizontal Asymptotes

Since the degree of the polynomial in the denominator is larger than the degree of the polynomial in the numerator, we have a horizontal asymptote at \(y=0\).

Find the Intercepts

The y-intercept is found by setting x to zero in the function, hence \(f(0) = -\frac{1}{2}\). This is the point (0,-0.5). The x-intercept is found by setting the function equal to zero and solving for x, but this function never equals zero so there are no x-intercepts.

Plot the Function

Having found all necessary information (asymptotes, intercepts), the function can now be sketched, keeping in mind that the graph will never cross the asymptotes.

Key concepts

Vertical Asymptotes

Vertical asymptotes in the graph of a rational function are found where the function is undefined, which occurs when the denominator equals zero. These vertical lines represent the x-values where the graph of the function shoots up to infinity or down to negative infinity.

To find the vertical asymptotes for the function \(f(x)=\frac{1}{x^2-x-2}\), we factor the denominator to \((x-2)(x+1)\). The zeroes of the denominator, \(x = 2\) and \(x = -1\), are the vertical asymptotes of the function. The graph is not allowed to touch or cross these lines at any point.

Horizontal Asymptotes

Horizontal asymptotes show the behavior of a graph as the x-values go towards positive or negative infinity. For the given function \(f(x)=\frac{1}{x^2-x-2}\), we note that the degree of the denominator is higher than that of the numerator.

This difference in degree means that as x grows larger in magnitude, the function's value approaches zero. Therefore, the horizontal asymptote for this function is the line \(y = 0\), which is also the x-axis. This line represents the value that the function's output is getting closer to, but will never actually reach, as the input (x) becomes very large or very small.

Intercepts

Intercepts are where a graph crosses the axes, providing initial points for sketching the graph. The y-intercept occurs where the graph of the function crosses the y-axis, which happens when x equals zero. For our function \(f(x)=\frac{1}{x^2-x-2}\), when we plug in \(x = 0\), we find the y-intercept at the point (0, -0.5).

The x-intercepts are found where the function equals zero, but since our function has a numerator of 1, it will never be equal to zero; hence, it has no x-intercepts. Knowing where the function intercepts the axes is crucial for plotting the initial points on the graph.

Factorization of Polynomials

Factorization of polynomials is a crucial step in graphing rational functions, as it allows us to identify vertical asymptotes and possible intercepts. By factoring the quadratic polynomial \(x^2-x-2\), we rewrite it in the factored form as \((x-2)(x+1)\).

Factorization can be done through methods like finding common factors, applying the difference of squares rule, or using the quadratic formula. The successful factorization of the denominator's polynomial in our function \(f(x)\) led us to identify the vertical asymptotes easily, which are essential for sketching a precise graph of the rational function.