Problem 12 - Algebra 2

Question

Describe the vertical asymptotes and holes for the graph of each rational function. $$ y=\frac{x+3}{(2 x+3)(x-1)} $$

Step-by-Step Solution

Answer

The vertical asymptotes of the function \(y=\frac{x+3}{(2x+3)(x-1)}\) are at \(x = -\frac{3}{2}\) and \(x = 1\). The function has no holes.

Step 1: Identifying Potential Asymptotes or Holes.

First, isolate the terms where the denominator could equal zero to identify potential vertical asymptotes: \(2x + 3 = 0\)\(x - 1 = 0\)

Step 2: Solving for x

Next, solve these equations separately to determine the x-values which would yield a denominator of zero: For \(2x + 3 = 0\), subtracting 3 from both sides and dividing by 2, gives \(x = -\frac{3}{2}\)For \(x - 1 = 0\), adding 1 to both sides gives \(x = 1\)

Step 3: Checking the Numerator

Now, review the function's numerator to see if it equals zero at either x = -3/2 or x = 1. If the numerator also equals zero at an x-value, this indicates a hole, not a vertical asymptote. Evaluating \(x + 3\), find that for x = -3/2, it does not equal zero, and for x = 1, it also does not equal zero.

Step 4: Distinguish Between Asymptotes and Holes

Since there are no x-values that make both the numerator and denominator zero, there are no holes in the function. So, the identified x-values of -3/2 and 1 are vertical asymptotes.

Key Concepts

Rational Functions

Rational functions are a fascinating area of algebra that involve the division of two polynomial expressions. Specifically, a rational function has the form \( f(x) = \frac{p(x)}{q(x)} \), where both \( p(x) \) and \( q(x) \) are polynomials. It's important to note that the denominator, \( q(x) \), should not be equal to zero, as division by zero is undefined. This is a crucial aspect as it dictates certain characteristics of the function like vertical asymptotes or holes.

In the function \( y = \frac{x+3}{(2x+3)(x-1)} \), \( x+3 \) is the numerator while \( (2x+3)(x-1) \) is the denominator. Understanding the behavior of these components helps predict how the graph behaves, such as where it might shoot off towards infinity or have missing points.

Graphs of Functions

Graphs of functions provide us with a visual representation of how a function behaves across different values of \( x \). For rational functions, this visualization becomes intriguing due to elements like vertical and horizontal asymptotes, and holes. These features can drastically change the shape and path of the graph, making it essential to pinpoint them while graphing.

• Vertical asymptotes occur where the denominator is zero and the numerator is not.
• Holes happen when both the numerator and denominator become zero at the same \( x \) value.

In our exercise, the graph of \( y = \frac{x+3}{(2x+3)(x-1)} \) will show vertical approaches to infinity near the asymptotes at \( x = -\frac{3}{2} \) and \( x = 1 \), where the denominator equals zero but the numerator does not.

Holes in Graphs

Holes in graphs of rational functions occur at specific points where both the numerator and the denominator evaluate to zero, effectively "canceling" each other out. This results in a "hole"—a point on the graph that is not defined.

To find holes, set both the numerator and denominator equal to zero; if a common \( x \) solution exists, a hole is present at that point. In the function \( y = \frac{x+3}{(2x+3)(x-1)} \), examining \( x+3 = 0 \) yields no matching zero with the denominator, hence no holes exist in this case.

Understanding this concept is essential for graphing as it identifies where the graph does not exist and ensures that calculations and interpretations of the function at this point are handled correctly.

Denominator Zero

The condition of having a denominator equal to zero in rational functions is significant. It leads to undefined points which manifest as vertical asymptotes or holes in the graph. Identifying these points involves examining the polynomial that is the denominator of the function like \( q(x) \).

• Set the denominator expressions equal to zero.
• Solve for \( x \) to find potential spots where the function is undefined.

For \( y = \frac{x+3}{(2x+3)(x-1)} \), solving \( 2x+3=0 \) and \( x-1=0 \) gives \( x=-\frac{3}{2} \) and \( x=1 \), respectively. These denote the vertical asymptotes since the numerator does not reach zero at these points. Recognizing these points helps one to avoid errors in calculations and to correctly describe the complete behavior of the function across its domain.