Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Sketch a graph of rational function. Your graph should include all asymptotes. Do not use a calculator. $$f(x)=\frac{x^{2}}{x^{2}-4}$$

Step-by-Step Solution

Answer

Graph has vertical asymptotes at \( x = -2, 2 \), horizontal asymptote \( y = 1 \), x-intercept at \( x = 0 \).

Step 1: Identify the Domain

To find the domain of the function \( f(x) = \frac{x^2}{x^2 - 4} \), set the denominator not equal to zero: \( x^2 - 4 eq 0 \). This simplifies to \( x^2 eq 4 \), solving gives \( x eq 2 \) and \( x eq -2 \). So, the domain is all real numbers except \( x = \pm 2 \).

Step 2: Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Since the function has factors \( x^2 - 4 = (x-2)(x+2) \), the vertical asymptotes occur at \( x = 2 \) and \( x = -2 \).

Step 3: Horizontal Asymptote

To find the horizontal asymptote, compare the degrees of the numerator and the denominator. Both are 2, so the horizontal asymptote is \( y = \frac{1}{1} = 1 \). This means as \( x \to \pm \infty \), \( f(x) \to 1 \).

Step 4: X-Intercepts

X-intercepts occur where the numerator is zero. For \( f(x) = \frac{x^2}{x^2 - 4} \), the numerator is zero at \( x = 0 \). So there is an x-intercept at \( x = 0 \).

Step 5: Y-Intercept

The y-intercept can be found by evaluating \( f(0) \). This gives \( f(0) = \frac{0^2}{0^2 - 4} = 0 \). Thus, the y-intercept is \( y = 0 \).

Step 6: Analyze Behavior Around Asymptotes

Examine the behavior of the function around the vertical asymptotes \( x = 2 \) and \( x = -2 \). As \( x \to 2^- \) or \( x \to 2^+ \), \( f(x) \to \pm \infty \). Similarly, as \( x \to -2^- \) or \( x \to -2^+ \), \( f(x) \to \pm \infty \). Check whether the function increases or decreases based on analysis around these points.

Step 7: Sketch the Graph

With all this information, sketch the graph noting the vertical asymptotes at \( x = -2 \) and \( x = 2 \) where the graph goes to \( \pm \infty \), the horizontal asymptote at \( y = 1 \), the x-intercept at \( (0,0) \), and approach towards \( y = 1 \) as \( x \to \pm \infty \).

Key Concepts

Domain of a Function

The domain of a rational function refers to all the possible values of the variable, usually referred to as "x", for which the function is defined. In the case of the function \( f(x) = \frac{x^2}{x^2 - 4} \), we determine the domain by identifying the values of "x" that will make the denominator zero, since division by zero is undefined in mathematics.

To find these critical points, set the denominator \( x^2 - 4 \) equal to zero: \( x^2 - 4 = 0 \). Solving this gives \( x^2 = 4 \), resulting in \( x = 2 \) and \( x = -2 \). These are the values where the function fails to be defined because they make the denominator zero. Therefore, the domain of \( f\) is all real numbers except \( x = \pm 2 \).

This can be expressed in interval notation as:

• \( (-\infty, -2) \cup (-2, 2) \cup (2, \infty) \)

Vertical Asymptotes

Vertical asymptotes are vertical lines on the graph of a rational function where the function approaches infinity or negative infinity. These occur at the values of "x" which make the denominator zero, and where the numerator does not simultaneously become zero.

In \( f(x) = \frac{x^2}{x^2 - 4} \), factorizing the denominator gives \( x^2 - 4 = (x-2)(x+2) \). The graph will have vertical asymptotes at \( x = 2 \) and \( x = -2 \) because these values make the denominator zero, thus the function tends towards infinity as it approaches these points.

Remember: A vertical asymptote means as "x" approaches the asymptote from either the left or the right, the function \( f(x) \) will go to \( +\infty \) or \( -\infty \).

The presence of vertical asymptotes suggests undefined points along the domain, requiring careful examination when analyzing the function's behavior or sketching its graph.

Horizontal Asymptotes

Horizontal asymptotes tell us about the behavior of a rational function as "x" approaches positive or negative infinity. They indicate the value that \( f(x) \) approaches, but does not necessarily reach, as "x" becomes very large or very small.

For the function \( f(x) = \frac{x^2}{x^2 - 4} \), the degree (highest power of "x") of both the numerator and denominator is 2. When the degrees are equal, the horizontal asymptote can be found by dividing the leading coefficients of the numerator and denominator.

In our expression, the leading coefficient is 1 for both the numerator and the denominator. So the horizontal asymptote is \( y = \frac{1}{1} = 1 \).

This means as "x" approaches \( \pm \infty \), the function \( f(x) \) gets closer and closer to the line \( y = 1 \). It is important to note that the function may cross its horizontal asymptote, but at further extremes, it will always tend towards it.

Intercepts

Intercepts are where the graph of a function crosses either the x-axis or y-axis. For rational functions, intercepts provide key positions where the graph cuts through these axes.

X-intercept: This occurs where \( f(x) = 0 \). Since the function's numerator must be zero, set \( x^2 = 0 \) to find the x-intercept for \( f(x) = \frac{x^2}{x^2 - 4} \). Solving \( x^2 = 0 \) gives \( x = 0 \). Hence, the x-intercept is at \( (0, 0) \).

Y-intercept: To determine the y-intercept, we evaluate \( f(x) \) at \( x = 0 \). Substituting gives \( f(0) = \frac{0^2}{0^2 - 4} = 0 \). Therefore, the y-intercept also occurs at \( (0, 0) \).

In this function, the x-intercept and y-intercept coincide at the same point. This happens when the graph passes through the origin, indicating a point of symmetry or a specific behavior in a rational function.