Let R(x) =( x2 + x − 1)/(3x^2 +11x -4). In order to sketch the graph of y = R(x),

To sketch the graph of the rational function \( R(x) = \frac{x^2 + x - 1}{3x^2 + 11x - 4} \), we need to analyze several aspects of the function:

1. **Domain**: Determine where the function is defined by finding the values of \( x \) that make the denominator zero:
   \[
   3x^2 + 11x - 4 = 0
   \]
   Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
   \[
   x = \frac{-11 \pm \sqrt{11^2 - 4 \cdot 3 \cdot (-4)}}{2 \cdot 3} = \frac{-11 \pm \sqrt{121 + 48}}{6} = \frac{-11 \pm \sqrt{169}}{6} = \frac{-11 \pm 13}{6}
   \]
   This gives us the roots:
   \[
   x = \frac{2}{6} = \frac{1}{3}, \quad x = \frac{-24}{6} = -4
   \]
   Thus, the function is undefined at \( x = \frac{1}{3} \) and \( x = -4 \).

2. **Intercepts**:
   - **Y-intercept**: Set \( x = 0 \):
     \[
     R(0) = \frac{0^2 + 0 - 1}{3 \cdot 0^2 + 11 \cdot 0 - 4} = \frac{-1}{-4} = \frac{1}{4}
     \]
   - **X-intercepts**: Set the numerator equal to zero:
     \[
     x^2 + x - 1 = 0
     \]
     Using the quadratic formula:
     \[
     x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} = \frac{-1 \pm \sqrt{5}}{2}
     \]

3. **Asymptotes**:
   - **Vertical asymptotes** occur at the values where the denominator is zero:
     \( x = -4 \) and \( x = \frac{1}{3} \).
   - **Horizontal asymptote**: Since the degrees of the numerator and denominator are both 2, the horizontal asymptote is given by the ratio of the leading coefficients:
     \[
     y = \frac{1}{3}
     \]

4. **Behavior Near Asymptotes**:
   - Analyze the behavior of \( R(x) \) as \( x \) approaches the vertical asymptotes from the left and right.

5. **End Behavior**:
   As \( x \to \pm \infty \), \( R(x) \) approaches \( \frac{1}{3} \).

### Summary of Graph Features:
- **Vertical Asymptotes**: \( x = -4 \) and \( x = \frac{1}{3} \)
- **Horizontal Asymptote**: \( y = \frac{1}{3} \)
- **Y-intercept**: \( (0, \frac{1}{4}) \)
- **X-intercepts**: \( x = \frac{-1 + \sqrt{5}}{2} \) and \( x = \frac{-1 - \sqrt{5}}{2} \)

### Graphing Steps:
1. Plot the intercepts and asymptotes.
2. Sketch the behavior of the function in each interval determined by the asymptotes.
3. Ensure the function approaches the horizontal asymptote as \( x \) goes to infinity.

Would you like to delve into any specific aspect of this function, or do you have any questions? Here are some follow-up questions:

1. How do you find the derivative of \( R(x) \)?
2. Can you explain the significance of vertical and horizontal asymptotes?
3. What happens to \( R(x) \) as \( x \) approaches the asymptotes?
4. How would you graph this function using technology?
5. Can you provide a numerical example of evaluating \( R(x) \) at specific points? 

**Tip**: Always check for intercepts and asymptotes when sketching rational functions, as they provide key information about the graph's behavior.

Let R(x) =( x^2 + x − 1)/(3x^2 +11x -4). In order to sketch the graph of y = R(x),

Explore how to sketch the graph of the rational function R(x) = (x^2 + x - 1) / (3x^2 + 11x - 4). This guide covers the function's domain, intercepts, asymptotes, and behavior near vertical asymptotes, making it ideal for students in Grades 10-12.

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