Problem 55 - Algebra 2

Question

What are the asymptotes of the graph of \(y=\frac{10}{x-5} ?\) A. \(x=0, y=5\) B. \(x=5, y=0\) C. \(x=5, y=10\) D. \(x=10, y=5\)

Step-by-Step Solution

Answer

Therefore, the asymptotes of the graph of the function are \(x=5\) and \(y=0\). So, option B. \(x=5, y=0\) is the correct answer.

Step 1: Identify the Form of the Function

The function \(y=\frac{10}{x-5}\) is in the form \(\frac{a}{x-h}\) where \(a=10\) and \(h=5\).

Step 2: Find the Vertical Asymptote

The vertical asymptote of a function of the form \(\frac{a}{x-h}\) is \(x=h\). In this case, that's \(x=5\).

Step 3: Find the Horizontal Asymptote

The horizontal asymptote of a function of the form \(\frac{a}{x-h}\) is \(y=0\).

Key Concepts

Vertical Asymptotes

Vertical asymptotes are vertical lines that represent the values for which a rational function becomes undefined. These occur in a rational function when its denominator is equal to zero, making the function undefined at those points. Imagine trying to divide a number by zero, it's impossible, thus, the graph cannot cross these lines.

For the function given in the exercise, \( y = \frac{10}{x-5} \), the denominator \( x-5 \) becomes zero when \( x = 5 \). Therefore, \( x = 5 \) is the vertical asymptote. This vertical line tells us that as \( x \) approaches 5, the function \( y \) will shoot up to positive or negative infinity, but will never actually touch or cross \( x = 5 \).

These lines are especially crucial because they tell us where the function will break or zoom off the chart.

Horizontal Asymptotes

Horizontal asymptotes are a bit different from vertical asymptotes. They tell us what value a rational function approaches as \( x \) either goes to positive infinity or negative infinity. Think of them as the levels at which the graph will settle when \( x \) becomes very large or very small.

In rational functions with the form \( y = \frac{a}{x-h} \), like \( y = \frac{10}{x-5} \) given in the exercise, the horizontal asymptote is a simple line at \( y = 0 \). This means as \( x \) continues to grow, the value of \( y \) will get closer and closer to zero but never actually reach it.

Horizontal asymptotes help us understand the behavior of the function at its extremes, offering a big picture of the graph's direction.

Rational Functions

Rational functions are at the core of understanding these asymptotes. A rational function is any function that can be expressed as the quotient of two polynomials, where the denominator is not zero.

A typical form of a rational function you might encounter is \( f(x) = \frac{a}{x-h} + k \), where \( a \), \( h \), and \( k \) are constants. These functions often contain features such as vertical and horizontal asymptotes, further guiding how their graphs look and behave.

For example, the function \( y = \frac{10}{x-5} \) in the exercise is a rational function. It demonstrates essential properties of these functions: it has a vertical asymptote at \( x = 5 \) because \( x = 5 \) would make the denominator zero. It also has a horizontal asymptote at \( y = 0 \) as \( x \) approaches infinity, showcasing how the function levels out.

By understanding these components, solving these types of exercises becomes much more approachable, letting you predict the graph's behavior with ease.