Problem 56 - College Algebra and Calculus: An Applied Approach

Question

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

Step-by-Step Solution

Answer

The function \(f(x)=\frac{2x}{x^{2}+x-2}\) has its x-intercept at \(x=0\), there is no y-intercept, the vertical asymptotes are at \(x=1\) and \(x=-2\), and the horizontal asymptote is at \(y=0\). The function has no symmetry.

Step 1: Check for Intercepts

To find the x-intercepts, set \(f(x)\) equal to zero and solve for \(x\). If \(f(x)=\frac{2x}{x^{2}+x-2}=0\), the numerator must be zero which gives \(x=0\). Thus, \(x=0\) is the x-intercept of the function. For the y-intercept, set \(x=0\) in the function. This would give an undefined value as the denominator will be -2, so there is no y-intercept.

Step 2: Check for Symmetry

To check if a function is even, replace \(x\) with \(-x\); if the function is unchanged, it's even. In this case, replacing \(x\) with \(-x\) produces a different function, so it's not even. To check if a function is odd, replace \(x\) with \(-x\); if the original function is returned negated, then the function is odd. In this case, no negation of original function is gained, meaning this function is neither even nor odd, thus it does not have symmetry about y-axis or about the origin.

Step 3: Check for Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero and numerator is not. For the function \(f(x)=\frac{2x}{x^{2}+x-2}\), set the denominator \(x^{2}+x-2 = 0\) and solve for \(x\). The roots are \(x_1 = 1\) and \(x_2 = -2\), hence \(x = 1\) and \(x = -2\) are the vertical asymptotes.

Step 4: Check for Horizontal Asymptotes

Horizontal asymptotes are determined by the degree and leading coefficients of the numerator and denominator. In this case, as the degree of the denominator is higher than the numerator, the horizontal asymptote is \(y = 0\).

Step 5: Plotting the Function

Plot the x and y intercepts, vertical and horizontal asymptotes, then fill in the graph so that it approaches and adheres to the intercepts and asymptotes identified. By following these steps an approximate sketch of the graph of function can be made.

Key Concepts

Understanding X-Intercepts

X-intercepts in a graph signify the points where a function crosses the x-axis. In the context of graphing rational functions like the given function
\( f(x) = \frac{2x}{x^2 + x - 2} \), we identify x-intercepts by setting the function equal to zero. To find the intercepts algebraically, you must solve for when the numerator is equal to zero.

This is due to the fact that a fraction equals zero only if its numerator is zero. In our example, setting \( 2x = 0 \) gives us the x-intercept \( x = 0 \). Importantly, if the function had more complex factors in the numerator, we'd solve for each to find multiple x-intercepts.

Finding Y-Intercepts

Y-intercepts are points where the function touches or crosses the y-axis, representing the function's value when \( x = 0 \). To find the y-intercept, you simply substitute zero for \( x \) in the function and solve for \( f(x) \).

In our problem, trying to evaluate \( f(0) \) results in division by zero, which is undefined. Hence, the given function does not have a y-intercept. Remember, a y-intercept isn't guaranteed; as seen here, certain functions may not touch the y-axis at all.

Vertical Asymptotes Essentials

Vertical asymptotes occur at points in the domain of a rational function where the function heads to positive or negative infinity. For the equation \( f(x) = \frac{2x}{x^2 + x - 2} \), we identify vertical asymptotes by setting the denominator equal to zero and solving for \( x \), as these would be the values where the function is not defined.

In this case, solving \( x^2 + x - 2 = 0 \) yields \( x = 1 \) and \( x = -2 \) as points where vertical asymptotes occur. This is critical for graphing as the curve approaches but never crosses these vertical lines.

Horizontal Asymptotes Simplified

Horizontal asymptotes provide a view of how a rational function behaves at extreme values of \( x \)—far to the left or the right of the graph. Determination of horizontal asymptotes depends on the degrees of the polynomials in the numerator and denominator.

Looking at our function, since the degree of the denominator (2) is greater than the degree of the numerator (1), we conclude that the horizontal asymptote is \( y = 0 \), which is the x-axis. Essentially, as \( x \) grows very large or very small, the function's value will get closer and closer to zero but never actually reach it.

Function Symmetry Analysis

Symmetry in functions can make graphing easier by allowing predictions of behavior on one side of a graph based upon the other. There are two primary types of symmetrical functions: even and odd. Even functions are symmetrical about the y-axis, whereas odd functions have origin symmetry.

Upon analysis, the given function \( f(x) \) does not exhibit symmetry. Replacing \( x \) with \( -x \) neither leaves the function unchanged (which would indicate even symmetry) nor results in its negation (which would indicate odd symmetry). Understanding symmetry helps in predicting and sketching the behavior of functions, but in this case, we must rely on other methods as the function \( f(x) = \frac{2x}{x^2 + x - 2} \) has no symmetry.