Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\frac{x+4}{x^{2}-4} $$

Step-by-Step Solution

Answer

Vertical asymptotes at \( x = 2 \) and \( x = -2 \); horizontal asymptote at \( y = 0 \); intercepts at \((0,-1)\) and \((-4,0)\).

Step 1: Identify the Domain

The domain of the function is all values of \( x \) for which the function is defined. Since there are denominators in the expression for \( f(x) = \frac{x+4}{x^2-4} \), we must avoid values that make the denominator zero. The expression \( x^2-4 \) factors into \( (x-2)(x+2) \), thus the denominator is zero when \( x = 2 \) or \( x = -2 \). Thus, the domain of \( f \) is all real numbers except \( x = 2 \) and \( x = -2 \).

Step 2: Find Asymptotes

Vertical asymptotes occur where the denominator is zero and the numerator is not. Therefore, there are vertical asymptotes at \( x = 2 \) and \( x = -2 \). For a horizontal asymptote, examine the degrees of the numerator and denominator. Since the degree of the denominator (2) is greater than the degree of the numerator (1), there is a horizontal asymptote at \( y = 0 \).

Step 3: Find Intercepts

To find the y-intercept, set \( x = 0 \): \( f(0) = \frac{0+4}{0^2-4} = \frac{4}{-4} = -1 \). So, the y-intercept is \((0, -1)\). For x-intercepts, set \( f(x) = 0 \) which gives the solution to \( x+4 = 0 \), thus \( x = -4 \). So the x-intercept is \((-4, 0)\).

Step 4: Consider Symmetry

Since \( f(-x) = \frac{-x+4}{(-x)^2-4} = \frac{-x+4}{x^2-4}\), \( f(x) \) is not even or odd, indicating no symmetry about the y-axis or origin.

Step 5: Sketch the Graph

On a graph, mark the intercepts: \((0, -1)\) and \((-4, 0)\). Draw vertical dashed lines at \( x = 2 \) and \( x = -2 \) for vertical asymptotes, and a horizontal dashed line along \( y = 0 \) for horizontal asymptote. Plot points on either side of each asymptote and connect them smoothly, getting closer to the asymptotes as you go to positive and negative infinity.

Key Concepts

Vertical Asymptotes

Vertical asymptotes occur in functions when the value of the denominator equals zero, making the function undefined at those specific points. In a rational function like \( f(x) = \frac{x+4}{x^2-4} \), the denominator \( x^2 - 4 \) can be factored into \((x-2)(x+2)\). This setup shows that vertical asymptotes appear at \( x = 2 \) and \( x = -2 \). Recognizing these vertical lines is crucial, as the function will approach them but never touch or cross them. When sketching, use dashed lines to mark these points on the graph, characterizing the undefined behavior precisely along these verticals.

Horizontal Asymptotes

Horizontal asymptotes guide us on how a function behaves as the variable \( x \) approaches infinity or negative infinity. The primary rule is this: when the degree of the polynomial in the denominator is greater than that of the numerator, as it is in \( f(x) = \frac{x+4}{x^2-4} \) (degree 2 vs. degree 1), the horizontal asymptote is at \( y = 0 \). This asymptote reflects that as \( x \) grows larger, the function approaches \( y = 0 \) and gets infinitely closer but never actually reaches it. The function levels out along this line, highlighting its long-term behavior.

Domain of a Function

Determining the domain of a function involves identifying all permissible inputs for \( x \) where the function still generates real outputs. The domain is often restricted by points where the denominator is zero. For \( f(x) = \frac{x+4}{x^2-4} \), the denominator becomes zero at \( x = 2 \) and \( x = -2 \), meaning the function is not defined at these values. Therefore, the domain is all real numbers excluding \( x = 2 \) and \( x = -2 \).

• The function is defined for all other values.
• Exclusions in the domain are clearly marked by vertical asymptotes.

Understanding the domain helps avoid calculation errors when dealing with rational functions.

Intercepts

Intercepts enlighten us about where the function crosses the coordinate axes. The y-intercept can be found by setting \( x = 0 \). For \( f(x) = \frac{x+4}{x^2-4} \), substituting \( x = 0 \) gives: \( f(0) = -1 \), translating into the graph crossing the y-axis at \( (0, -1) \).
The x-intercept is determined by \( f(x) = 0 \), equating to solving \( x+4 = 0 \). Hence, the x-intercept is \( (-4, 0) \). At this point, the graph touches or crosses the x-axis. Knowing both intercepts helps in accurately sketching the graph's initial layout.

Graph Symmetry

Graph symmetry provides insight into the function's visual layout, checking for reflective properties about specific axes or the origin.
To assess symmetry, substitute \( -x \) into \( f(x) \):
\( f(-x) = \frac{-x+4}{(-x)^2-4} = \frac{-x+4}{x^2-4} \).
This result shows no equality to \( f(x) \) or \( -f(x) \), indicating the function has neither y-axis symmetry nor origin symmetry.

• The absence of symmetry means the graph does not mirror around the y-axis or resemble a centralized rotation symmetry about the origin.
• This knowledge aids in anticipating the graph's overall shape.

Analyzing symmetry is vital for a comprehensive understanding of the graph's behavior.