Graph the rational function. 6 -2x-1 Start by drawing the vertical and horizontal asymptotes. Then f(x) =
Graph the rational function. 6 -2x-1 Start by drawing the vertical and horizontal asymptotes. Then f(x) =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Graphing Rational Functions
In this lesson, we will learn how to graph a rational function. Let’s consider the following rational function:
\[ f(x) = \frac{-6}{2x - 1} \]
#### Step-by-Step Instructions:
1. **Identify the Asymptotes:**
- **Vertical Asymptote:** The vertical asymptote occurs where the denominator is zero. Set the denominator equal to zero and solve for \( x \):
\[ 2x - 1 = 0 \]
\[ 2x = 1 \]
\[ x = \frac{1}{2} \]
So, the vertical asymptote is at \( x = \frac{1}{2} \).
- **Horizontal Asymptote:** To determine the horizontal asymptote, we observe the degrees of the polynomial in the numerator and the denominator. In this case, both the numerator and the denominator are of the same degree:
\[ \frac{\text{Coefficient of } x^1 \text{ in the numerator}}{\text{Coefficient of } x^1 \text{ in the denominator}} = \frac{-6}{2} = -3 \]
So, the horizontal asymptote is \( y = -3 \).
2. **Plotting the Asymptotes on the Graph:**
- Draw a dashed vertical line at \( x = \frac{1}{2} \).
- Draw a dashed horizontal line at \( y = -3 \).
3. **Plot Points to Define the Function Behavior:**
- A few points can be calculated to further understand the graph's shape, but this is not shown in the provided image.
4. **Draw the Rational Function:**
- The rational function \( f(x) = \frac{-6}{2x - 1} \) can now be graphed. It will approach but never touch the vertical line at \( x = \frac{1}{2} \) and the horizontal line at \( y = -3 \).
#### Graph
Below is the graph of the function \( f(x) = \frac{-6}{2x - 1} \).
[Image of a Cartesian coordinate system with grid lines. The vertical asymptote is represented by a dashed line at \( x = \frac{1}{2} \), and the horizontal asym](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbae73a37-719e-4892-8745-1f816a181ea7%2F24d53350-d2b5-4f68-ad2e-9ff495bfebdd%2Fwt9wbsd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Graphing Rational Functions
In this lesson, we will learn how to graph a rational function. Let’s consider the following rational function:
\[ f(x) = \frac{-6}{2x - 1} \]
#### Step-by-Step Instructions:
1. **Identify the Asymptotes:**
- **Vertical Asymptote:** The vertical asymptote occurs where the denominator is zero. Set the denominator equal to zero and solve for \( x \):
\[ 2x - 1 = 0 \]
\[ 2x = 1 \]
\[ x = \frac{1}{2} \]
So, the vertical asymptote is at \( x = \frac{1}{2} \).
- **Horizontal Asymptote:** To determine the horizontal asymptote, we observe the degrees of the polynomial in the numerator and the denominator. In this case, both the numerator and the denominator are of the same degree:
\[ \frac{\text{Coefficient of } x^1 \text{ in the numerator}}{\text{Coefficient of } x^1 \text{ in the denominator}} = \frac{-6}{2} = -3 \]
So, the horizontal asymptote is \( y = -3 \).
2. **Plotting the Asymptotes on the Graph:**
- Draw a dashed vertical line at \( x = \frac{1}{2} \).
- Draw a dashed horizontal line at \( y = -3 \).
3. **Plot Points to Define the Function Behavior:**
- A few points can be calculated to further understand the graph's shape, but this is not shown in the provided image.
4. **Draw the Rational Function:**
- The rational function \( f(x) = \frac{-6}{2x - 1} \) can now be graphed. It will approach but never touch the vertical line at \( x = \frac{1}{2} \) and the horizontal line at \( y = -3 \).
#### Graph
Below is the graph of the function \( f(x) = \frac{-6}{2x - 1} \).
[Image of a Cartesian coordinate system with grid lines. The vertical asymptote is represented by a dashed line at \( x = \frac{1}{2} \), and the horizontal asym
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