Problem 82 - Algebra and Trigonometry

Question

A. Find the slant asymplote of the graph of each rational function and B. Follow the seven-step strategy and use the slant asymptote to graph each rational function. $$ f(x)=\frac{x^{2}-4}{x} $$

Step-by-Step Solution

Answer

The slant asymptote of the function \(f(x)=\frac{x^{2}-4}{x}\) is \(y=x\).

Step 1: Identify the highest degree of the polynomial

For the function \(f(x)=\frac{x^{2}-4}{x}\), the highest degree of the polynomial is 2, which is in the numerator.

Step 2: Perform Polynomial Division

Divide the numerator by the denominator. We can express \(x^{2}-4\) as \(x*x-4\). Consider x as the divisor and divide \(x*x\), the first term in the dividend, which gives us x. Multiply the divisor x by the quotient x to get \(x^2\). Subtract \(x^2\) from \(x^2\) (first term in the dividend), which results in 0. Bring down the next term in the divisor -4 to get -4. Now, \(x\) cannot divide -4 as \(x^2 > -4\), so we stop the division. The quotient obtained is the equation of the slant asymptote.

Step 3: Find the Slant Asymptote

The quotient obtained from the division in Step 2 is x, so the equation of the slant asymptote is \(y=x\).

Step 4: Graph the function and the slant asymptote

Plot the slant asymptote \(y = x\) on the graph. Now, graph the function \(f(x)=\frac{x^{2}-4}{x}\). You will notice that the function approaches the slant asymptote as \(x\to\infty\) or \(x \to -\infty\) but will never reach the slant asymptote.

Key Concepts

Rational Functions

Rational functions are an important type of function in mathematics, characterized by the form \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) eq 0 \). These functions can take various shapes depending on the degree of the polynomials in the numerator and the denominator.
In this exercise, the function \( f(x)=\frac{x^{2}-4}{x} \) is a rational function. The numerator is a polynomial \( x^2 - 4 \) and the denominator is \( x \). Such functions are often studied for their asymptotic behavior and graph characteristics.
Key features of rational functions include:

• Vertical asymptotes, which occur where the denominator is zero.
• Horizontal or slant asymptotes, which describe the behavior of the function as \( x \) approaches infinity.
• X-intercepts and y-intercepts, which are found by setting the numerator and denominator to zero, respectively.

Understanding these features helps to graph rational functions and predict their long-term behavior.

Polynomial Division

Polynomial division is a technique used to simplify rational functions, especially when finding asymptotes. In this process, you divide the polynomial in the numerator by the one in the denominator, similar to long division with numbers.
For the function \( f(x)=\frac{x^{2}-4}{x} \), we perform polynomial division to find the slant asymptote. The numerator \( x^2 - 4 \), when divided by the denominator \( x \), results in a quotient of \( x \). This is because:

• First, divide the leading term of the numerator \( x^2 \) by the leading term of the denominator \( x \) to get \( x \).
• Then, multiply \( x \) by \( x \) to subtract \( x^2 \) from the numerator, leaving \( -4 \) in the process.
• Since \(-4\) cannot be further divided by \( x \), we stop, and our quotient \( x \) is the slant asymptote.

The slant asymptote, \( y = x \), indicates that as \( x \) goes towards infinity or negative infinity, the function \( f(x) \) will approach this line.

Graphing Functions

Graphing functions like \( f(x)=\frac{x^{2}-4}{x} \) involves plotting key components such as asymptotes, intercepts, and the general shape of the function on a coordinate plane. Understanding these elements before sketching the graph can make the process simpler and more accurate.
The seven-step strategy for graphing involves:

• Identifying the domain.
• Finding intercepts.
• Determining vertical asymptotes by setting \( q(x) = 0 \).
• Finding horizontal or slant asymptotes using polynomial division if needed.
• Plotting a table of values to further understand the function’s behavior.
• Identifying any symmetries or end behaviors.
• Combining all these steps to draw the function and its asymptotes.

For the given function, plot the slant asymptote \( y = x \) and observe that as \( x \) approaches positive or negative infinity, the graph of \( f(x) \) will get closer to this line. Remember, the function never actually touches the slant asymptote, but merely approaches it.