Problem 1 - Algebra for College Students

Question

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{x^{2}}{x^{2}+x-2} $$

Step-by-Step Solution

Answer

The graph has vertical asymptotes at \(x = -2\) and \(x = 1\), and a horizontal asymptote at \(y = 1\); it passes through \((0, 0)\).

Step 1: Check for Symmetry

To check for symmetry, determine if the function is even, odd, or neither. Recall:- A function is even if \(f(-x) = f(x)\).- A function is odd if \(f(-x) = -f(x)\).Calculate \(f(-x)\): \(f(-x) = \frac{(-x)^2}{(-x)^2 + (-x) - 2} = \frac{x^2}{x^2 - x - 2}\).Since \(f(x)\) is not equal to \(f(-x)\) or \(-f(-x)\), the function is neither even nor odd. Therefore, it's not symmetric.

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator is not zero.Find the zeros of the denominator: \(x^2 + x - 2 = 0\).Factor the quadratic: \((x + 2)(x - 1) = 0\).The solutions are \(x = -2\) and \(x = 1\).Thus, the vertical asymptotes are at \(x = -2\) and \(x = 1\).

Step 3: Identify Horizontal Asymptotes

Horizontal asymptotes are determined by the degrees of the numerator and denominator:- If the degree of the numerator (2) equals the degree of the denominator (2), the horizontal asymptote is the ratio of leading coefficients.The leading coefficients are both 1, so the horizontal asymptote is \(y = \frac{1}{1} = 1\).

Step 4: Check Intercepts

Find the x-intercepts by setting the numerator equal to zero: \(x^2 = 0\) gives \(x = 0\).Therefore, the x-intercept is \((0, 0)\).Find the y-intercept by evaluating \(f(0)\): \(f(0) = \frac{0^2}{0^2 + 0 - 2} = 0\).This confirms the y-intercept is also \((0, 0)\).

Step 5: Graph the Function

Plot the vertical asymptotes at \(x = -2\) and \(x = 1\), and the horizontal asymptote at \(y = 1\).Plot the intercept \((0, 0)\).Draw the behavior of the graph, considering it will approach but not touch the asymptotes. Ensure the vertical asymptotes divide the x-axis into intervals and check the function behavior in each interval (such as \(x < -2\), \(-2 < x < 1\), and \(x > 1\)).

Key Concepts

Symmetry in Functions

When analyzing a rational function for symmetry, we check if the function is even, odd, or neither. A function is considered even if replacing

• \(x\) with \(-x\) yields the same function: \(f(-x) = f(x)\).
• A function is odd if this substitution results in the negative of the original function: \(f(-x) = -f(x)\).

For our given function \(f(x) = \frac{x^2}{x^2 + x - 2}\), let's find \(f(-x)\): \[f(-x) = \frac{(-x)^2}{(-x)^2 + (-x) - 2} = \frac{x^2}{x^2 - x - 2}\] Since neither \(f(-x) = f(x)\) nor \(f(-x) = -f(x)\), this indicates that the function is neither even nor odd. As a result, there is no symmetry about the y-axis or origin. Recognizing symmetry can simplify graphing tasks, but in this case, we must proceed without it.

Vertical Asymptotes

Vertical asymptotes of a rational function arise from values that make the denominator zero while leaving the numerator non-zero. So, we begin by setting the denominator of \(f(x)\) to equal zero: \[x^2 + x - 2 = 0\] Given the quadratic, we can factor it as \((x + 2)(x - 1) = 0\), revealing zeros at \(x = -2\) and \(x = 1\). These zeros indicate that vertical asymptotes exist at these x-values. Hence, the graph of the function will approach these lines vertically, and the function will be undefined where \(x = -2\) and \(x = 1\). Vertical asymptotes divide the x-axis into distinct intervals, each of which may behave differently. Checking the behavior in intervals such as \(x < -2\), \(-2 < x < 1\), and \(x > 1\), helps sketch the graph accurately.

Horizontal Asymptotes

Horizontal asymptotes provide insight into the long-term behavior of a rational function as \(x\) approaches infinity or negative infinity. For the function \(f(x) = \frac{x^2}{x^2 + x - 2}\), we compare the degrees of the numerator and the denominator.

• If the degree of the numerator (2) equals the degree of the denominator (2), the horizontal asymptote is given by the ratio of their leading coefficients.

In this instance, both the numerator and the denominator have a degree of 2, with leading coefficients being 1. Thus, the horizontal asymptote is at: \[y = \frac{1}{1} = 1\] This informs us that as \(x\) becomes very large or very small, the function will settle towards \(y=1\). Understanding where horizontal asymptotes lie helps predict how the graph will behave at its extremes.

Intercepts in Graphs

Intercepts are crucial for understanding where a graph crosses the x-axis or y-axis. To find the x-intercepts, set the numerator of the rational function to zero:\[x^2 = 0\]Solving yields \(x = 0\), which implies the x-intercept is at the point \((0, 0)\). Similarly, the y-intercept is found by evaluating \(f(0)\):\[f(0) = \frac{0^2}{0^2 + 0 - 2} = 0\] Thus, the function's y-intercept is also \((0, 0)\). These intercepts indicate where the function graph crosses the axes, which are essential in plotting and visualizing the function accurately.Graphically, these points can assist in anchoring the function as it approaches or recedes from asymptotes.