Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\frac{-3 x^{2}-3 x+6}{x^{2}-9} $$

Step-by-Step Solution

Answer

The graph has vertical asymptotes at \(x = 3\) and \(x = -3\), a horizontal asymptote at \(y = -3\), and a y-intercept at \((0, -\frac{2}{3})\).

Step 1: Identify Domain

To find the domain of the function, set the denominator equal to zero and solve: \( x^2 - 9 = 0 \). This gives \( x = \pm 3 \). Thus, the domain of \( f(x) \) is all real numbers except \( x = -3 \) and \( x = 3 \).

Step 2: Find Vertical Asymptotes

Vertical asymptotes occur when the denominator approaches zero but the numerator does not. In \( f(x) \), set \( x^2 - 9 = 0 \), which occurs at \( x = 3 \) and \( x = -3 \). Since the numerator does not vanish at \( x = 3 \) and \( x = -3 \), vertical asymptotes exist at these points.

Step 3: Determine Horizontal Asymptote

The degrees of the numerator and the denominator are both 2. Thus, the horizontal asymptote is \( y = \frac{-3}{1} = -3 \), which is the ratio of the leading coefficients.

Step 4: Calculate Intercepts

Set \( f(x) = 0 \) to find the x-intercepts. Solve \(-3x^2 - 3x + 6 = 0\). The x-intercepts are solutions to this equation. Also, set \( x = 0 \) to find the y-intercept: \( f(0) = \frac{6}{-9} = -\frac{2}{3} \).

Step 5: Analyze Graph Behavior

Evaluate the function's behavior near the vertical asymptotes by approaching from the left and the right sides. Also, analyze near the horizontal asymptote as \( x \to \pm \infty \). Understand any possible crossing over.

Step 6: Sketch the Graph

Using the asymptotes, intercepts, and behavior calculated, sketch the graph showing the appropriate directions and intervals based on the domain and intercepts.

Key Concepts

Domain of a Function

The domain of a function represents all the possible input values (usually x-values) for which the function is defined. To determine the domain of a rational function like \( f(x) = \frac{-3x^2 - 3x + 6}{x^2 - 9} \), we must identify the values that make the denominator zero, as division by zero is undefined.

In this case, set the denominator equal to zero: \( x^2 - 9 = 0 \). Solving this equation, we get \( x = 3 \) and \( x = -3 \). These values are excluded from the domain.

Therefore, the domain of the function \( f \) is all real numbers except \( x = 3 \) and \( x = -3 \). You can write this as \( \{ x \mid x eq 3, x eq -3 \} \). By knowing the domain, we ensure we only use valid x-values for analyzing or graphing the function.

Vertical Asymptotes

Vertical asymptotes are lines that the graph of a function approaches but never touches or crosses. These typically occur in rational functions where the denominator approaches zero, but the numerator does not.

For \( f(x) = \frac{-3x^2 - 3x + 6}{x^2 - 9} \), we find the vertical asymptotes by setting the denominator to zero: \( x^2 - 9 = 0 \), resulting in \( x = 3 \) and \( x = -3 \). Since the numerator \(-3x^2 - 3x + 6\) does not become zero for these x-values, vertical asymptotes exist at \( x = 3 \) and \( x = -3 \).

Understanding vertical asymptotes helps us anticipate the behavior of the function as x approaches these critical points. On a graph, you can see the function will rise or fall sharply as it approaches the asymptote lines.

Horizontal Asymptotes

Horizontal asymptotes show the behavior of a function as the input values become very large or very small (approach infinity). They are particularly relevant for rational functions where both the numerator and denominator are polynomials.

In our function \( f(x) = \frac{-3x^2 - 3x + 6}{x^2 - 9} \), both the numerator and denominator are quadratic polynomials (degree 2). The horizontal asymptote in this case is determined by the ratio of the leading coefficients of the numerator and the denominator.

Here, the leading coefficient of the numerator is \(-3\), and for the denominator, it is \(1\). Thus, the horizontal asymptote is \( y = \frac{-3}{1} = -3 \). This means that as \( x \to \pm \infty \), the function approaches the line \( y = -3 \). Horizontal asymptotes give us a visual marker for the long-term behavior of the graph as x moves towards large positive or negative values.

Intercepts of a Function

Intercepts are where a graph crosses the axes, specifically the x-axis and the y-axis. These points can provide quick, important information about a function.

The x-intercepts occur where \( f(x) = 0 \). For our function \( f(x) = \frac{-3x^2 - 3x + 6}{x^2 - 9} \), solve the equation \(-3x^2 - 3x + 6 = 0\) to find the x-intercepts. Solving this quadratic gives points where the function crosses the x-axis.

The y-intercept occurs where \( x = 0 \). Substitute 0 for \( x \) in the function: \( f(0) = \frac{6}{-9} = -\frac{2}{3} \). This indicates that the graph crosses the y-axis at \( y = -\frac{2}{3} \).

Identifying intercepts is particularly useful for sketching the graph as they define some of the exact points the function will definitely pass through.