Problem 43 - BIG IDEAS MATH Algebra 2: Common Core Student Edition 2015

Question

What is the vertical asymptote of the graph of the function \(y=\frac{2}{x+4}+7\) ? (A) \(x=-7\) (B) \(x=-4\) (C) \(x=4\) (D) \(x=7\)

Step-by-Step Solution

Answer

The vertical asymptote of the given function is \(x=-4\). So, option (B) is the correct answer.

Step 1: Identify the Function

The given function is the rational function \(y=\frac{2}{x+4}+7\). The denominator of the rational part of the function is \(x+4\).

Step 2: Find the Value That Makes the Denominator Zero

Set the denominator equal to zero and solve for \(x\), \(x+4=0\) gives \(x=-4\). This means that the given function will be undefined at \(x=-4\).

Step 3: Determine the Vertical Asymptote

Vertical asymptote is the line \(x = a\) where the function is undefined. Since the function is undefined at \(x=-4\), the vertical asymptote of the graph of this function is \(x=-4\).

Key Concepts

Rational Functions

Rational functions are mathematical expressions representing the division of two polynomials. The general form of a rational function is \( f(x) = \frac{P(x)}{Q(x)} \), where both \( P(x) \) and \( Q(x) \) are polynomials and \( Q(x) \) is not the zero polynomial. An essential characteristic of rational functions is their behavior near certain values of \( x \) that cause \( Q(x) \) to be zero. These specific values are significant as they can lead to vertical asymptotes in the graph of the function.

Rational functions often model real-world situations where ratios are involved, such as speed, density, and concentration. Understanding how these functions work and how to graph them is crucial for analyzing such scenarios effectively. When graphing rational functions, it is-important to determine the points where the function is undefined, which leads us to the concept of 'undefined function values'.

Undefined Function Values

In mathematics, there are instances where a function does not have a defined value for certain inputs. For rational functions, this occurs when the denominator is zero since division by zero is undefined. These specific input values, which result in an undefined function, directly relate to the concept of vertical asymptotes.

In our example \( y=\frac{2}{x+4}+7 \), the function becomes undefined when \( x+4 = 0 \), which happens at \( x=-4 \). At this point, the value of \( y \) cannot be determined, signifying a discontinuity in the graph. Hence, \( x=-4 \) is excluded from the domain of the function. Recognizing these undefined values is a key step in solving equations involving rational functions, as it helps delineate the function's domain and identify potential vertical asymptotes.

Solving Equations

Solving equations is a fundamental skill in algebra that requires finding the values that satisfy the equality. With rational functions, the process typically involves setting the equation equal to zero and solving for the variable. This technique is especially useful in identifying vertical asymptotes because these asymptotes are found at the values of \( x \) where the function's denominator equals zero, leading to undefined function values.

To solve the equation for our exercise, \( x+4=0 \), we simply subtract 4 from both sides to isolate \( x \) on one side, resulting in \( x=-4 \). This solution indicates that the function does not exist at this particular \( x \)-value and consequently, a vertical asymptote is present here. Understanding this process is integral to analyzing and graphing rational functions and recognizing the larger implications of undefined values within the function's domain.