Problem 52 - 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. $$f(x)=\frac{x}{4-x^{2}}$$

Step-by-Step Solution

Answer

Asymptotes at x = 2, x = -2; intercept at (0,0); horizontal asymptote y = 0. The function changes sign across asymptotes.

Step 1: Identify the asymptotes

To locate vertical asymptotes, look for the values of \(x\) that make the denominator zero. For the function \(f(x) = \frac{x}{4-x^2}\), set the denominator equal to zero: \[ 4 - x^2 = 0. \]Solving for \(x\), we get \[ x^2 = 4 \]\[ x = \pm 2. \]Thus, there are vertical asymptotes at \(x = 2\) and \(x = -2\). The horizontal asymptote is determined by the degrees of the numerator and denominator polynomials. Because both have the same degree (1 in the numerator and 2 in the denominator), the horizontal asymptote is \(y = 0\).

Step 2: Find the intercepts

To find the y-intercept, evaluate the function at \(x = 0\):\[ f(0) = \frac{0}{4-0^2} = 0. \]So, the y-intercept is at \((0,0)\). Since the numerator will be zero when \(x=0\), the x-intercept is also at \((0,0)\).

Step 3: Analyze the behavior near asymptotes

Analyze the sign of \(f(x)\) as \(x\) approaches the vertical asymptotes \(x = 2\) and \(x = -2\). Check the behavior from the left and right of these points:- As \(x\) approaches 2 from the left (e.g., \(x \to 2^-\)), \(x^2 < 4\), so \(4 - x^2 > 0\) and \(f(x) > 0\).- As \(x\) approaches 2 from the right (e.g., \(x \to 2^+\)), \(x^2 > 4\), so \(4 - x^2 < 0\) and \(f(x) < 0\).- Similarly, at \(-2^+\) and \(-2^-\), evaluate to see the function mirrors these behaviors. This implies there is a sign change across each asymptote.

Step 4: Sketch the graph

Start the graph by marking key points: the asymptotes at \(x = 2\), \(x = -2\), and \(y = 0\), and the intercept at \((0,0)\). The function has opposite signs on different sides of the vertical asymptotes. Knowing this, you predict the behavior:- From \(x = -\infty\) to \(-2\), \(f(x)\) is in the positive region.- Between \(-2\) and \(0\), \(f(x)\) is negative.- Between \(0\) and \(2\), \(f(x)\) returns positive.- From \(2\) to \(+\infty\), \(f(x)\) is negative. Plot the behavior, keeping in mind the function approaches \(y = 0\) as \(x\to \pm \infty\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are lines where the graph of a function tends to infinity in either the positive or negative direction as the input (usually noted as \( x \)) approaches certain values. For rational functions like \( f(x) = \frac{x}{4-x^2} \), these asymptotes occur where the denominator equals zero. In our function, this means solving \( 4 - x^2 = 0 \). By rearranging and solving for \( x \), we find \( x^2 = 4 \), which gives \( x = \pm 2 \). So, the vertical asymptotes are at \( x = 2 \) and \( x = -2 \).

• Remember, the graph cannot cross vertical asymptotes. They represent values where the function is undefined.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as \( x \) approaches positive or negative infinity. They show the value that \( f(x) \) approaches. For rational functions, horizontal asymptotes depend on the degrees of the polynomial in the numerator and the denominator.

• If the degree of the denominator is greater than the degree of the numerator, as in \( f(x) = \frac{x}{4-x^2} \) where the denominator's degree is 2 and the numerator's is 1, the horizontal asymptote is \( y = 0 \).

As \( x \) increases or decreases without bound, the graph approaches this line but may never actually touch it. The horizontal line \( y = 0 \) is where the graph settles at infinity.

Intercepts

Intercepts are points where the graph crosses the axes. **Finding Intercepts:**

• The y-intercept occurs where the graph intersects the y-axis, meaning \( x = 0 \).
• For our function, setting \( x = 0 \) means \( f(0) = \frac{0}{4-0^2} = 0 \). So, the y-intercept is at \( (0,0) \).
• Conveniently, because \( x \) makes the numerator of a rational function zero, this is also the x-intercept for \( f(x) \).

Intercepts provide useful anchor points for sketching the graph as they are guaranteed to be on the graph itself.

Graph Sketching

Graph sketching involves using all the information from the asymptotes and intercepts to draw an accurate depiction of the function's behavior. First, plot the vertical asymptotes at \( x = 2 \) and \( x = -2 \) as dashed lines. These represent the boundaries where the function grows towards infinity. Then, draw the horizontal asymptote at \( y = 0 \), understanding that the graph flattens out along this line as \( x \) tends to infinity.

• Place the intercept at \( (0,0) \), where the graph crosses the origin.
• Check the function's sign on either side of the vertical asymptotes to determine the direction:
• For \( x < -2 \), \( f(x) > 0 \).
• Between \( -2 < x < 0\), \( f(x) < 0 \).
• From \( 0 < x < 2 \), \( f(x) > 0 \) again.
• For \( x > 2 \), \( f(x) < 0 \).

Use these insights to draw a smooth curve that hugs the asymptotes and crosses through the intercepts. Always recheck the sign changes to avoid errors in drawing Polly the Assumed Witch.