Problem 98 - College Algebra

Question

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{2 x-9}{x-4}$$

Step-by-Step Solution

Answer

The equation transformed is \(g(x)=2 + \frac{-1}{x-4}\). The transformations are horizontal shift 4 units to the right, vertical shift 2 units up, and reflection across the x-axis.

Step 1: Long Division of Polynomials

Performing long division on \((2x - 9)\) by \((x - 4)\), the quotient is \(2\) and the remainder is \(-1\). This can be summarized as follows: Start by dividing the leading terms to get the first term of the quotient. \(\frac{2x}{x}\) equals \(2\). Place this above the line. Multiply the entire divisor, \((x - 4)\), by this quotient term, \(2\), and subtract the result from the original polynomial \((2x - 9)\). This gives a new polynomial \(-1\). Since the degree of this polynomial is less than the degree of the divisor, this is the remainder of the division.

Step 2: Rewrite the Equation

From the long division, we can now rewrite the equation in the form quotient + \(\frac{remainder}{divisor}\). Therefore, the function \(g(x)=\frac{(2x-9)}{(x-4)}\) can be transformed into \(g(x)=2 + \frac{-1}{x-4}\).

Step 3: Identify the Transformations

The graph of the function \(g(x)=2 + \frac{-1}{x-4}\) is a transformation of the graph of \(y=\frac{1}{x}\). It includes the following transformations: horizontal shift 4 units to the right, vertical shift of 2 units upward, and reflection across the x-axis.

Key Concepts

Polynomial Division

Polynomial division is a technique that helps us break down complex rational expressions into simpler forms. When you have a polynomial divided by another polynomial, this process is similar to long division with numbers.
To perform polynomial division:

• Identify the leading term of both the dividend (the numerator) and the divisor (the denominator).
• Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
• Multiply the entire divisor by this new term and subtract the result from the dividend. This will give you the remainder.
• Repeat these steps with the new polynomial formed (remainder) until the degree of the remainder is less than the degree of the divisor.

In the given exercise, dividing \((2x - 9)\) by \((x - 4)\) gives a quotient of \(2\) and a remainder of \(-1\), simplifying our initial polynomial expression.

Rational Functions

Rational functions are one of the main types of expressions involving polynomials. They can be represented as the ratio of two polynomial functions. These functions often have behaviors that create asymptotes in their graphs, which are lines the graph approaches but never touches.
For a rational function given as \(\frac{A(x)}{B(x)}\), where \(A(x)\) and \(B(x)\) are polynomials, key points include:

• Vertical asymptotes occur where \(B(x) = 0\), indicating the denominator is zero.
• Horizontal asymptotes are often determined by comparing the degrees of the numerator and the denominator.
• The overall behavior of the function depends on transformations, shifts, and reflections influenced by these factors.

In our exercise, \(g(x) = \frac{2x - 9}{x - 4}\), is a rational function initially, but through division and simplification, becomes a more approachable expression for further transformations.

Function Transformations

Understanding function transformations can be a powerful tool in visualizing and analyzing the behavior of graphs. Transformations adjust the position, shape, and size of the graph of the function in a predictable manner.
Some common types of transformations include:

• Horizontal shifts move the graph left or right. For example, \(x-4\) in the denominator suggests a shift of 4 units to the right.
• Vertical shifts move the graph up or down. The term \(+2\) moves the graph 2 units upward.
• Reflections involve flipping the graph over an axis. If the function is \(-\), it indicates a reflection across the x-axis.

Our simplified expression, \(g(x) = 2 + \frac{-1}{x-4}\), illustrates these concepts:- The function is shifted right by 4 units, flipped across the x-axis, and shifted 2 units upward.
These transformations give us a clear view on where the function moves, helping us understand how the graph behaves.