Math Problem Statement

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

Solution

To sketch the graph of the rational function R(x)=x2+x13x2+11x4R(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 xx that make the denominator zero: 3x2+11x4=03x^2 + 11x - 4 = 0 Using the quadratic formula x=b±b24ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}: x=11±11243(4)23=11±121+486=11±1696=11±136x = \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=26=13,x=246=4x = \frac{2}{6} = \frac{1}{3}, \quad x = \frac{-24}{6} = -4 Thus, the function is undefined at x=13x = \frac{1}{3} and x=4x = -4.

  2. Intercepts:

    • Y-intercept: Set x=0x = 0: R(0)=02+01302+1104=14=14R(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: x2+x1=0x^2 + x - 1 = 0 Using the quadratic formula: x=1±1241(1)21=1±52x = \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=4x = -4 and x=13x = \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=13y = \frac{1}{3}
  4. Behavior Near Asymptotes:

    • Analyze the behavior of R(x)R(x) as xx approaches the vertical asymptotes from the left and right.
  5. End Behavior: As x±x \to \pm \infty, R(x)R(x) approaches 13\frac{1}{3}.

Summary of Graph Features:

  • Vertical Asymptotes: x=4x = -4 and x=13x = \frac{1}{3}
  • Horizontal Asymptote: y=13y = \frac{1}{3}
  • Y-intercept: (0,14)(0, \frac{1}{4})
  • X-intercepts: x=1+52x = \frac{-1 + \sqrt{5}}{2} and x=152x = \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 xx 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)R(x)?
  2. Can you explain the significance of vertical and horizontal asymptotes?
  3. What happens to R(x)R(x) as xx approaches the asymptotes?
  4. How would you graph this function using technology?
  5. Can you provide a numerical example of evaluating R(x)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.

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Math Problem Analysis

Mathematical Concepts

Rational Functions
Graphing
Algebra

Formulas

Quadratic formula: x = (-b ± √(b² - 4ac)) / 2a
Horizontal asymptote: y = leading coefficient of numerator / leading coefficient of denominator

Theorems

Asymptote Theorems
Properties of Rational Functions

Suitable Grade Level

Grades 10-12