Question

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{2+x}{1-x}$$

Short answer

The graph of the function \(f(x) = \frac{2+x}{1-x}\) has an x-intercept at (-2,0), y-intercept at (0,2), a vertical asymptote at \(x=1\) and a horizontal asymptote at \(y=-1\) .

Step-by-step solution

Find Intercepts

To find the \(x\)-intercept, make \(f(x) = 0\). \nGiven \(f(x)=\frac{2+x}{1-x}\), when set to equal to 0, it implies that the numerator is 0. Meaning \(2 + x = 0\), we find \(x = -2\). The \(y\)-intercept is found by substituting \(x = 0\) into \(f(x)\). After substitution, we get \(f(0) = 2\). Therefore, the axis intercepts are (-2,0) and (0,2).

Check for Symmetry

Rational functions of this form (\(f(x) = \frac{P(x)}{Q(x)}\), with \(P(x)\) and \(Q(x)\) polynomials) are not generally symmetric; they are not same on both sides of the y-axis, neither do they reflect about the origin. Therefore, it is not symmetric.

Find Vertical Asymptotes

Vertical asymptotes occur when the denominator equals zero, since division by zero is undefined. In our case, setting \(1-x = 0\) gives \(x=1\). Therefore, \(x=1\) is the vertical asymptote.

Find Horizontal Asymptotes

Horizontal asymptotes reflect the behavior of the function as \(x\) approaches infinity or negative infinity. It's determined by examining the degrees of the numerator and denominator. If the degree of the numerator is less than the denominator, the x-axis is the horizontal asymptote. If their degrees are equal, divide the leading coefficients. If the degree of the numerator is greater, there's no horizontal asymptote. Here, the degrees of the numerator and denominator are both equal to 1, hence we divide the leading coefficients, yielding \(-1\). Therefore, the line \(y=-1\) is our horizontal asymptote.

Sketch the Graph

Start by drawing the vertical line \(x=1\) and horizontal line \(y=-1\) to indicate the asymptotes. Plot the intercepts at (-2,0) and (0,2). Since the function is not symmetric, attempt to take a few points on either side of the vertical asymptote. For \(x>1\), try \(x=2\), then \(f(2) = -4\). For \(x<1\), use \(x=0\), then \(f(0) = 2\). Using these points and accounting for the asymptotic behavior, one can sketch a reasonable graph.

Key concepts

Vertical Asymptotes

Vertical asymptotes are like invisible barriers through which the graph of a rational function cannot pass. They represent the values of x for which the function is undefined, usually because the denominator of the rational function is zero at those points. For example, in the function

\[ f(x)=\frac{2+x}{1-x} \]

the denominator is zero when x equals 1. Therefore, there's a vertical asymptote at x=1. When graphing, this asymptote appears as a dashed vertical line, and it helps us predict the behavior of the function near that value. It's important not to confuse this with a hole in the graph, which occurs if the same factor exists in both the numerator and denominator and they cancel each other out. In our function, no such cancellation occurs, thus confirming x=1 as a true vertical asymptote.

Horizontal Asymptotes

Horizontal asymptotes serve as a guide to where the graph of a function stretches or compresses as x approaches infinity or negative infinity. These lines represent the limit to which the function values will approach but never actually reach.

In the given function, \[ f(x) = \frac{2+x}{1-x} \], the degrees of the numerator and denominator are equal, thus we look at the coefficients of the highest power terms of x. In this case, both are 1, but since the x in the denominator has a negative sign, our horizontal asymptote is where the leading coefficients ratio is -1, which gives us the line y=-1. When graphing, this helps students anticipate that regardless of the large values, positive or negative, that x takes, the function's values get closer to -1 but will never touch it.

X-Intercepts and Y-Intercepts

The points where the graph of a function crosses the x-axis and y-axis are known as the x-intercepts and y-intercepts, respectively. These are the points of intersection with the axes, revealing where the function has an output of zero, or in the case of the y-intercept, where the function passes through when the input x is zero.

For the rational function, \[ f(x)=\frac{2+x}{1-x} \], the x-intercept is found by setting the function equal to zero. This gives us the point (-2,0), after solving for when the numerator equals zero. The y-intercept is found by evaluating the function at x=0, which yields the point (0,2). Plotting these intercepts on the graph gives a clearer picture of how the function behaves near the origin and provides key points around which the graph is anchored.

Asymptotic Behavior

Asymptotic behavior is a fundamental concept when graphing rational functions as it describes how the function behaves at the extreme ends of the graph or near the asymptotes. As x values grow large in both the positive and negative direction, or as they approach specific values causing the function to be undefined, the function's graph will tend to get closer to a line without ever touching it—this line is the asymptote.

In the case of the function \[ f(x)=\frac{2+x}{1-x} \], we have already identified vertical and horizontal asymptotes at x=1 and y=-1, respectively. By taking additional points near these asymptotes, we can observe that the graph will approach these lines but not cross them, defining the curvature and direction of the graph. The behavior near the vertical asymptote is particularly interesting as the value of f(x) becomes more and more extreme, flirting with infinity as x approaches 1 from the left and negative infinity as it approaches from the right. Understanding asymptotic behavior is crucial for accurately sketching the overall shape of a rational function's graph.