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.