Problem 52 - College Algebra

Question

For the following exercises, write an equation for a rational function with the given characteristics. Vertical asymptotes at \(x=-4\) and \(x=-1, x\) -intercepts at \((1,0)\) and \((5,0), y\) -intercept at \((0,7)\)

Step-by-Step Solution

Answer

The function is \(f(x) = \frac{7(x - 1)(x - 5)}{(x + 4)(x + 1)}\)."

Step 1: Understand the Vertical Asymptotes

Vertical asymptotes occur where the denominator of a rational function is zero. Given vertical asymptotes at \(x = -4\) and \(x = -1\), we know the denominator must include the factors \((x + 4)\) and \((x + 1)\). Therefore, part of the denominator can be expressed as \((x + 4)(x + 1)\).

Step 2: Determine the x-intercepts

The x-intercepts occur where the numerator of the rational function is zero. Given x-intercepts at \((1, 0)\) and \((5, 0)\), the numerator must include the factors \((x - 1)\) and \((x - 5)\). Thus, part of the numerator can be expressed as \((x - 1)(x - 5)\).

Step 3: Incorporate the y-intercept

The y-intercept tells us the value of the function when \(x = 0\). We need the function to equal 7 when \(x = 0\). Initialize the function \(f(x)\) as: \(f(x) = \frac{a(x - 1)(x - 5)}{(x + 4)(x + 1)}\). Substitute \(x = 0\) and \(f(0) = 7\) to find \(a\): \[7 = \frac{a(0 - 1)(0 - 5)}{(0 + 4)(0 + 1)}\].

Step 4: Solve for the multiplier "a"

Plug in the values and simplify the equation from step 3: \[7 = \frac{a(-1)(-5)}{4}\], solving \(35 = 5a\), gives \(a = 7\).

Step 5: Write the Final Rational Function

Using \(a = 7\) in the function from Step 3, we finalize the function: \(f(x) = \frac{7(x - 1)(x - 5)}{(x + 4)(x + 1)}\). This equation satisfies all the given conditions of vertical asymptotes at \(x = -4, -1\), x-intercepts at \((1, 0), (5, 0)\), and a y-intercept at \((0, 7)\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes in a rational function occur where the function does not have a value ─ often referred to as points of undefined behavior. These appear when the denominator of the function equals zero. To find the vertical asymptotes, simply look at the factors of the denominator. For instance, if your function has a denominator of

• \((x + 4)(x + 1)\)

setting these factors to zero will help us find the vertical asymptotes.

• Solve \(x + 4 = 0\) which gives \(x = -4\)
• Solve \(x + 1 = 0\) which gives \(x = -1\)

Thus, the vertical asymptotes are located at \(x = -4\) and \(x = -1\). Whenever the function approaches these x-values, it will go towards infinity or negative infinity, reflecting these vertical lines.

X-Intercepts

An x-intercept is a point where the graph of a function crosses the x-axis. This occurs when the output of the function is zero, meaning the numerator of the rational function must be zero with a non-zero denominator. Using the equation's characteristics, our x-intercepts are given at

• \((1,0)\)
• \((5,0)\)

This tells us that the numerator should include factors such as

• \((x - 1)\)
• \((x - 5)\)

By solving each factor for zero,

• The intercept at \((1,0)\) occurs because \(x - 1 = 0\), giving \(x = 1\)
• The intercept at \((5,0)\) occurs because \(x - 5 = 0\), giving \(x = 5\)

Hence, these locations are crucial when designing the numerator for any rational function meant to match these x-intercepts.

Y-Intercepts

Finding the y-intercept of a rational function means determining where the graph crosses the y-axis. This happens when \(x = 0\). It's essentially finding the output of the function at this particular value of \(x\). Given the y-intercept at

• \((0,7)\)

we need to ensure our function adequately reflects this point. The function can be formulated as:\[ f(x) = \frac{a(x - 1)(x - 5)}{(x + 4)(x + 1)} \]Setting \(x = 0\) and solving for \(f(0) = 7\) provides the necessary multiplier \(a\). This ensures that at \(x = 0\), the function's value is 7, aligning with the required y-intercept on the graph.

Denominator and Numerator of Rational Functions

Rational functions are like fractions, with a numerator and a denominator, both of which are polynomials. The structure and factors of these two polynomials determine the various intercepts and asymptotes seen in the graph. In the exercise example, the rational function was intended to have

• Vertical asymptotes at \(x = -4\) and \(x = -1\)
• X-intercepts at \((1,0)\) and \((5,0)\)
• A y-intercept at \((0,7)\)

To satisfy these conditions, the function takes the following form:\[ f(x) = \frac{7(x - 1)(x - 5)}{(x + 4)(x + 1)} \]

• The numerator \(7(x-1)(x-5)\) is crafted to hit the x-intercepts at \( (1, 0) \) and \( (5, 0) \).
• The denominator \((x + 4)(x + 1)\) ensures the vertical asymptotes are correctly placed.
• The leading constant 7 guarantees the correct y-intercept.

By putting together the right factors for both the numerator and the denominator, the desired behavior of the function graph is elegantly achieved.