Problem 54 - College Algebra

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 rational function is \(f(x) = \frac{-2(x + 2)(x - 1)}{(x + 3)(x - 6)}\).

Step 1: Identify the structure for vertical asymptotes

Vertical asymptotes occur when the denominator is zero but the numerator is non-zero at those points. Since there are vertical asymptotes at \(x = -3\) and \(x = 6\), the function will have factors \((x + 3)\) and \((x - 6)\) in the denominator.

Step 2: Identify the structure for x-intercepts

X-intercepts occur where the numerator is zero. Given \(x\)-intercepts at \((-2, 0)\) and \((1, 0)\), the function will have factors \((x + 2)\) and \((x - 1)\) in the numerator.

Step 3: Determine the horizontal asymptote condition

Horizontal asymptote at \(y = -2\) suggests that the degrees of the numerator and denominator polynomials are equal. Additionally, the ratio of the leading coefficients must be \(-2\). Thus, the leading coefficient of the numerator should be \(-2\) if the leading coefficient of the denominator is \(1\). This means our function starts as \(-2(x + 2)(x - 1)\) over \((x + 3)(x - 6)\).

Step 4: Construct the rational function

Combine the elements from previous steps: the complete rational function is given by \[f(x) = \frac{-2(x + 2)(x - 1)}{(x + 3)(x - 6)}\]. This function satisfies all the specified conditions: vertical asymptotes at \(x = -3, 6\), \(x\)-intercepts at \((-2, 0)\) and \((1, 0)\), and a horizontal asymptote at \(y = -2\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes give us important information about the behavior of a rational function. They occur when approaching certain values of \(x\) causes the function to shoot off towards infinity, either positive or negative. This is because the denominator of the rational function becomes zero at that point, while the numerator is non-zero.

For example, in our function, there are vertical asymptotes at \(x = -3\) and \(x = 6\). To have a vertical asymptote at a particular \(x\)-value, the function must include corresponding factors in the denominator that become zero when \(x\) takes on these values:

• At \(x = -3\), the factor is \((x + 3)\).
• At \(x = 6\), the factor is \((x - 6)\).

This behavior showcases the importance of vertical asymptotes in determining how the function behaves around those \(x\)-values, allowing us to predict and understand the limits of the function's graphs.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as \(x\) approaches positive or negative infinity. They suggest a value that the function levels out towards in its extremes. Understanding horizontal asymptotes helps us comprehend how the function behaves far outside the range of typical \(x\)-values.

For instance, the given function features a horizontal asymptote at \(y = -2\). This indicates that as \(x\) becomes very large in either direction, the values of the function approach \(-2\). This happens when the degrees of the numerator and denominator polynomials are equal, reflecting the ratio of the leading coefficients:

• The leading coefficient of the numerator must be \(-2\) if the denominator's leading coefficient is \(1\).

In our function, both the numerator and the denominator are of degree 2. Therefore, the horizontal asymptote is accurately represented by the ratio \(-2/1\), which simplifies to \(y = -2\). This elegant characteristic helps us visualize the stability of the rational function's output in its extremities.

X-intercepts

X-intercepts are crucial for finding where a rational function crosses the x-axis. A function will have x-intercepts where the numerator is zero and the denominator is non-zero. These are the points where the output of the function equals zero.

In our rational function example, we have x-intercepts provided at \((-2, 0)\) and \((1, 0)\). The numerator must include factors that become zero at these specific \(x\)-values:

• At \(x = -2\), the factor is \((x + 2)\).
• At \(x = 1\), the factor is \((x - 1)\).

These x-intercepts help define the function's graph on the x-axis. They are essential in plotting and understanding how the curve interacts with the x-axis, showing exactly where the function gets to zero output. By identifying these intercepts, we gain clearer insight into the roots of the rational function.