Problem 54 - College Algebra with Corequisite Support

Question

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptotes at \(x=-3\) and \(x=6\), \(x\) -intercepts at (-2,0) and \((1,0),\) Horizontal asymptote at \(y=-2\)

Step-by-Step Solution

Answer

The equation is \(f(x) = -2\frac{(x + 2)(x - 1)}{(x + 3)(x - 6)}\)."

Step 1: Identify the Components of the Rational Function

The rational function will have vertical asymptotes where the denominator is zero. Since the vertical asymptotes are at \(x = -3\) and \(x = 6\), the denominator must include the factors \((x + 3)(x - 6)\). The \(x\)-intercepts are at \((-2, 0)\) and \((1, 0)\), meaning the numerator must include the factors \((x + 2)(x - 1)\). Lastly, the horizontal asymptote is at \(y = -2\), which implies the degrees of the numerator and the denominator must be equal, and the leading coefficient of the numerator must be \(-2\).

Step 2: Construct the Numerator and Denominator

Given the \(x\)-intercepts, the numerator of the function is \((x + 2)(x - 1)\). Given the vertical asymptotes, the denominator is \((x + 3)(x - 6)\). To ensure the horizontal asymptote at \(y = -2\), multiply the entire numerator by \(-2\). Thus, the function becomes \(f(x) = -2\frac{(x + 2)(x - 1)}{(x + 3)(x - 6)}\).

Step 3: Write the Equation of the Rational Function

Now substitute the components found in the previous steps into the rational function format: \(f(x) = -2\frac{(x + 2)(x - 1)}{(x + 3)(x - 6)}\).This function has vertical asymptotes at \(x = -3\) and \(x = 6\), \(x\)-intercepts at \(x = -2\) and \(x = 1\), and a horizontal asymptote at \(y = -2\).

Key Concepts

Understanding Vertical Asymptotes

Vertical asymptotes are a key feature of rational functions. They represent values of \( x \) where the graph of the function will stretch indefinitely towards positive or negative infinity. In simpler terms, the graph will approach these vertical lines but will never actually touch them. Vertical asymptotes occur when the denominator of the rational function is zero, as division by zero is undefined. Thus, in the expression \( f(x) = -2\frac{(x + 2)(x - 1)}{(x + 3)(x - 6)} \), the vertical asymptotes are identified at the values of \( x \) that make the denominator \( (x + 3)(x - 6) \) equal to zero: namely \( x = -3 \) and \( x = 6 \). To find vertical asymptotes, remember to:

• Set the denominator equal to zero.
• Solve for \( x \).

Vertical asymptotes provide crucial insights into the behavior of a rational function around these critical points.

Exploring Horizontal Asymptotes

Horizontal asymptotes are another important aspect of rational functions. They describe the behavior of a function as \( x \) approaches positive or negative infinity. Essentially, these asymptotes give us an idea of the value that the function will tend toward as its input becomes very large or very small. For our function, the horizontal asymptote is given as \( y = -2 \). This horizontal line becomes the value that the function approaches but never quite reaches at extreme values of \( x \). In the case of rational functions where the degrees of the numerator and denominator are the same, the horizontal asymptote is found by dividing the leading coefficients of these two parts. Here, both the numerator \( (x + 2)(x - 1) \) and the denominator \( (x + 3)(x - 6) \) have degree 2. Thus, the horizontal asymptote is given by the coefficient \( -2 \) from the function\'s formula:

• If the degrees of the numerator and denominator are equal, divide the leading coefficients.
• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).
• If the degree of the numerator is greater, there is no horizontal asymptote (but there might be an oblique asymptote).

Understanding horizontal asymptotes helps predict the end behavior of a rational function's graph.

Determining X-Intercepts

The \( x \)-intercepts of a rational function occur at the points where the graph crosses the x-axis. Unlike vertical asymptotes, these involve setting the numerator of the rational function to zero, because when a function equals zero, its graph intersects the x-axis. For the function \( f(x) = -2\frac{(x + 2)(x - 1)}{(x + 3)(x - 6)} \), the \( x \)-intercepts are found by setting the numerator \( (x + 2)(x - 1) \) equal to zero, leading to solutions \( x = -2 \) and \( x = 1 \).

• Set the numerator equal to zero.
• Solve for \( x \).
• These solutions reveal the \( x \)-intercepts.

Identifying \( x \)-intercepts is crucial for understanding where the graph of the rational function will actually touch or cross the x-axis, offering a clearer picture of its overall shape and intersections.