Problem 4 - Algebra for College Students

Question

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{-x^{2}}{x^{2}+3 x-4} $$

Step-by-Step Solution

Answer

The function has vertical asymptotes at \( x = -4 \) and \( x = 1 \), and a horizontal asymptote at \( y = -1 \); it has no symmetry.

Step 1: Determine Symmetry

To check for symmetry, replace \( x \) with \( -x \) in the function: \[ f(-x) = \frac{-(-x)^2}{(-x)^2 + 3(-x) - 4} = \frac{-x^2}{x^2 - 3x - 4} \]The function \( f(-x) eq f(x) \) and \( f(-x) eq -f(x) \), so there is no symmetry about the y-axis, the origin, or the x-axis.

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur at values of \( x \) where the denominator equals zero. Solve the equation:\[ x^2 + 3x - 4 = 0 \]Factoring gives:\[ (x+4)(x-1) = 0 \]So, vertical asymptotes exist at \( x = -4 \) and \( x = 1 \).

Step 3: Identify Horizontal Asymptotes

Horizontal asymptotes depend on the degrees of the numerator and the denominator. Both are quadratic polynomials (degree 2), so the horizontal asymptote is given by the leading coefficients of \(-x^2\) and \(x^2\):\[ y = \frac{-1}{1} = -1 \]Thus, the horizontal asymptote is \( y = -1 \).

Step 4: Sketch the Graph

1. Draw the vertical asymptotes at \( x = -4 \) and \( x = 1 \), represented by dashed vertical lines. 2. Draw the horizontal asymptote at \( y = -1 \) as a dashed horizontal line.3. Since there are no symmetries, plot points to understand the behavior of the function, focusing on intervals defined by the asymptotes.4. Sketch the curve, showing it approaching the asymptotes but never crossing the vertical ones.

Key Concepts

Graphing Rational Functions

Graphing rational functions involves understanding the behavior of a function given in the form of a fraction, or ratio, of two polynomials. Such functions can have curves that behave intriguingly, especially near points where the denominator is zero. There are a few steps to follow when graphing these functions:

• Identify asymptotes, which are lines that the graph approaches but never touches.
• Check for symmetry to simplify your graphing process.
• Find intercepts of the function if needed for additional points.
• Plot key points to help draw the curve.

By carefully arranging this information on a graph, you can accurately sketch the behavior of complex rational functions.

Asymptotes in Calculus

Asymptotes are lines that a graph approaches but never actually reaches. In calculus, they play a crucial role in understanding the end-behavior of functions. There are three main types of asymptotes:

• Vertical Asymptotes: Typically occur where the denominator of a rational function equals zero, causing undefined values.
• Horizontal Asymptotes: Occur based on the relative degrees of the numerator and denominator, indicating behavior as x approaches infinity.
• Oblique Asymptotes: Happen when the polynomial in the numerator is one degree higher than the polynomial in the denominator. However, they aren't present in our specific exercise problem.

Understanding these asymptotic behaviors aids in sketching more accurate graphs, demonstrating how functions behave at extreme values of x.

Function Symmetry

Function symmetry is an important concept for simplifying the graphing of rational functions. Symmetry can help identify predictable patterns in a graph, reducing the need for plotting numerous points.
Some possible symmetries include:

• Y-axis symmetry: If a function is unchanged when x is replaced with -x, it's said to be symmetric about the y-axis.
• Origin symmetry: If replacing x with -x also requires negating the entire function to yield the same function, symmetry exists about the origin.
• X-axis symmetry: Rare in functions, this occurs if negating the y-value still results in the graph of the function.

In our given exercise, replacing x with -x does not yield the original or negated function, indicating no symmetry.

Vertical Asymptotes

Vertical asymptotes are found by setting the denominator of the rational function to zero and solving for x. This provides points where the graph goes upwards or downwards indefinitely.

• To find them, solve the equation obtained by setting the denominator equal to zero.
• In our function, solving the quadratic equation \( x^2 + 3x - 4 = 0 \) leads to roots of \( x = -4 \) and \( x = 1 \), which are our vertical asymptotes.
• On the graph, these points are represented as dashed vertical lines where the function cannot have a value.

Offending points like these lead to undefined values and are essential in understanding the graph's behavior.

Horizontal Asymptotes

Horizontal asymptotes are derived from comparing the degrees of the polynomials in the numerator and the denominator of the rational function. They inform us of the end behavior of the function as x moves towards positive or negative infinity.

• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
• When both have the same degree, the horizontal asymptote is found by taking the ratio of the leading coefficients.
• If the numerator's degree is greater, no horizontal asymptote exists unless it's just one degree higher, in which an oblique asymptote might exist.

In our exercise, both the numerator and denominator have degree 2, leading to a horizontal asymptote at \( y = -1 \), calculated as the ratio of leading coefficients \( -1 : 1 \). This line is depicted as a horizontal dashed line on the graph.