Graph the rational function. f(x)= 5x+15 x+2x-15 2 Start by drawing the vertical and horizontal asymptotes. Finally, click on the graph-a-function button.
Graph the rational function. f(x)= 5x+15 x+2x-15 2 Start by drawing the vertical and horizontal asymptotes. Finally, click on the graph-a-function button.
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
#### Graph the Rational Function
\[ f(x) = \frac{5x + 15}{x^2 + 2x - 15} \]
**Step-by-Step Instructions:**
1. **Identify and Draw Asymptotes:**
- **Vertical Asymptotes:** These occur where the denominator \( x^2 + 2x - 15 \) is equal to zero. Solve for \( x \):
\[ x^2 + 2x - 15 = 0 \]
Factor the quadratic equation:
\[ (x + 5)(x - 3) = 0 \]
Therefore, \( x = -5 \) and \( x = 3 \) are the vertical asymptotes.
- **Horizontal Asymptotes:** As \( x \) approaches infinity, the horizontal asymptote is determined by comparing the degrees of the numerator and the denominator. Here, both degrees are the same and the horizontal asymptote occurs at:
\[ y = \frac{5}{1} = 5 \]
2. **Graph Preparation:**
- To accurately graph, draw the vertical asymptotes \( x = -5 \) and \( x = 3 \) as vertical dashed lines.
- Draw the horizontal asymptote \( y = 5 \) as a horizontal dashed line.
3. **Plot the Function:**
- Use the given graphing interface (suggested as “graph-a-function button”) to plot the function \( f(x) \).
- Ensure the graph covers the range on both the positive and negative sides of the asymptotes.
**Graph Description:**
The provided graph is a standard Cartesian plane with the x-axis and y-axis extending from -14 to 14. The graph includes grid lines at unit intervals to assist with plotting functions accurately. The goal is to represent the rational function \( f(x) \) within this grid, highlighting the behavior around the identified vertical and horizontal asymptotes.
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Transcribed Image Text:---
### Graphing Rational Functions
#### Graph the Rational Function
\[ f(x) = \frac{5x + 15}{x^2 + 2x - 15} \]
**Step-by-Step Instructions:**
1. **Identify and Draw Asymptotes:**
- **Vertical Asymptotes:** These occur where the denominator \( x^2 + 2x - 15 \) is equal to zero. Solve for \( x \):
\[ x^2 + 2x - 15 = 0 \]
Factor the quadratic equation:
\[ (x + 5)(x - 3) = 0 \]
Therefore, \( x = -5 \) and \( x = 3 \) are the vertical asymptotes.
- **Horizontal Asymptotes:** As \( x \) approaches infinity, the horizontal asymptote is determined by comparing the degrees of the numerator and the denominator. Here, both degrees are the same and the horizontal asymptote occurs at:
\[ y = \frac{5}{1} = 5 \]
2. **Graph Preparation:**
- To accurately graph, draw the vertical asymptotes \( x = -5 \) and \( x = 3 \) as vertical dashed lines.
- Draw the horizontal asymptote \( y = 5 \) as a horizontal dashed line.
3. **Plot the Function:**
- Use the given graphing interface (suggested as “graph-a-function button”) to plot the function \( f(x) \).
- Ensure the graph covers the range on both the positive and negative sides of the asymptotes.
**Graph Description:**
The provided graph is a standard Cartesian plane with the x-axis and y-axis extending from -14 to 14. The graph includes grid lines at unit intervals to assist with plotting functions accurately. The goal is to represent the rational function \( f(x) \) within this grid, highlighting the behavior around the identified vertical and horizontal asymptotes.
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