Problem 27 - Algebra and Trigonometry with Analytic Geometry

Question

Sketch the graph of \(f\). $$ f(x)=\frac{-2 x^{2}+10 x-12}{x^{2}+x} $$

Step-by-Step Solution

Answer

The graph has vertical asymptotes at \( x=0 \), \( x=-1 \), a horizontal asymptote at \( y=-2 \), and x-intercepts at \( x=3 \), and \( x=2 \).

Step 1: Determine the Domain

To find the domain of the function \( f(x) = \frac{-2x^2 + 10x - 12}{x^2 + x} \), set the denominator equal to zero and solve for \( x \). The denominator is \( x^2 + x = 0 \), which factors as \( x(x+1) = 0 \). Therefore, the domain is all real numbers except \( x = 0 \) and \( x = -1 \).

Step 2: Find the Asymptotes

For vertical asymptotes, set the denominator \( x^2 + x = 0 \). Thus, vertical asymptotes are at \( x = 0 \) and \( x = -1 \). For horizontal asymptotes, compare the degree of the numerator \(-2x^2 + 10x - 12\) and the denominator \(x^2 + x\). Since they are the same, the horizontal asymptote is at \( y = \frac{-2}{1} = -2 \).

Step 3: Determine Intercepts

To find the y-intercept, evaluate \( f(0) \), but this leads to division by zero, so no y-intercept exists. Find x-intercepts by setting the numerator \(-2x^2 + 10x - 12 = 0\). Factoring gives \(-2(x-3)(x-2) = 0\). The x-intercepts are \( x = 3 \) and \( x = 2 \).

Step 4: Analyze Behavior Between Asymptotes and Intercepts

Check the behavior of the function around the asymptotes \( x = 0 \) and \( x = -1 \), as well as intervals between the intercepts. For example, check sample points to determine whether \( f(x) \) approaches \( \pm \infty \) or follows the asymptote (\( y= -2 \)).

Step 5: Sketch the Graph

Combine all gathered information: domain, intercepts, asymptotes, and behavior analysis. Draw the vertical asymptotes as dashed lines at \( x = 0 \) and \( x = -1 \), the horizontal line at \( y = -2 \), and plot the intercepts at \( (3,0) \) and \( (2,0) \). Sketch the curve approaching asymptotes and passing through intercepts.

Key Concepts

Domain of Rational Functions

The domain of rational functions refers to all possible x-values that a function can take without leading to mathematically undefined situations. Specifically, for a function \[ f(x) = \frac{P(x)}{Q(x)} \]we need to determine when the denominator, \( Q(x) \), is not equal to zero. This is because division by zero is undefined. In our exercise, the denominator is \( x^2 + x \). **Steps to find the domain:**

• Factor the denominator: \( x(x + 1) = 0 \).
• Solve the equation: \( x = 0 \) and \( x = -1 \). These values make the denominator zero, hence they are excluded from the domain.

Thus, the domain of the function is all real numbers except \( x = 0 \) and \( x = -1 \). Knowing the domain is essential to understand where the function is defined.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually touches or crosses. For rational functions, these are typically vertical or horizontal. To locate them, we use:**Vertical Asymptotes:**- Vertical asymptotes occur where the denominator is zero, as long as the numerator is not zero at these points too. From our factorization:

• Vertical asymptotes are at \( x = 0 \) and \( x = -1 \).

**Horizontal Asymptotes:**- To determine horizontal asymptotes, compare the degrees of the numerator and denominator. If both have the same degree:

• The horizontal asymptote is the ratio of the leading coefficients. Our function has a horizontal asymptote at \( y = -2 \), because the leading coefficients of the numerator and denominator are \(-2\) and \(1\), respectively.

These asymptotes guide us in understanding the overall structure and end behavior of the function.

Intercepts of Rational Functions

Intercepts are points where the graph of the function crosses the axes:**X-Intercepts:**- To find the x-intercepts, set the numerator equal to zero and solve for \( x \):

• \(-2x^2 + 10x - 12 = 0\). Factoring gives \(-2(x-3)(x-2) = 0\).
• The x-intercepts are \( x = 3 \) and \( x = 2 \).

**Y-Intercepts:**- The y-intercept is found by evaluating \( f(0) \), but in this case, since the denominator becomes zero at \( x=0 \), no y-intercept exists.Intercepts provide critical points through which the graph passes, helping to determine its precise shape and position.

Behavior Analysis of Graphs

Understanding the behavior of a graph involves examining how it behaves towards infinity and near asymptotes. This involves:**Analyzing Behavior Near Asymptotes:**- As we approach the vertical asymptotes at \( x = 0 \) and \( x = -1 \), the function can either head towards \( \infty \) or \(-\infty \). Evaluating points around these asymptotes helps determine the direction.**Behavior Between Intercepts:**- Check sample points in intervals divided by intercepts \( x = 2 \) and \( x = 3 \) to see if they lie above or below the horizontal asymptote \( y = -2 \).Such analysis gives insight into how the graph moves across its domain, whether it follows expected trends, and how it behaves near critical points like intercepts and asymptotes.