Analyzing Rational Functions

Rational Functions and Graphing

Overview of Rational Functions

  • Rational functions are unique as they can have holes in their graphs.

  • A hole in the graph occurs at specific points of the function when certain factors cancel out.

    • Definition: A hole is a point in the graph of a function where the function is not defined, typically due to the cancellation of terms in the numerator and denominator.

  • The identification of holes involves:

    • Determining when factors cancel out.

    • Identifying the zero of the canceled factor to derive the x-coordinate of the hole.

    • Plugging this x-value into the new function (derived after cancellation) to find the y-coordinate of the hole.

Steps to Identify Holes and Graph Rational Functions

  • The process includes four major tasks:

    1. Find vertical asymptotes (VA)

    2. Identify holes (if any)

    3. Determine x-intercepts

    4. Determine y-intercepts

Example 1: Finding Holes and Asymptotes

  • Given the function: f(x)=2x2−8x2+3x+2f(x) = \frac{2x^2 - 8}{x^2 + 3x + 2}

Factoring

  • Factor both numerator and denominator:

    • Numerator: 2(x+2)(x−2)2(x + 2)(x - 2)

    • Denominator: (x+1)(x+2)(x + 1)(x + 2)

  • Upon factoring, identify the canceled factor: (x+2)(x + 2).

Identifying Holes

  • x-coordinate of the hole:

    • Set the canceled factor to zero: x+2=0⇒x=−2x + 2 = 0 \Rightarrow x = -2

  • To find the y-coordinate, substitute x = -2 into the new function.

    • New function after cancelation: f(x)=2(x−2)(x+1)f(x) = \frac{2(x - 2)}{(x + 1)}

    • Plugging in: f(−2)=2(−2−2)(−2+1)=2(−4)−1=8f(-2) = \frac{2(-2 - 2)}{(-2 + 1)} = \frac{2(-4)}{-1} = 8

  • Therefore, the hole is at (-2, 8).

Vertical Asymptote

  • To find the vertical asymptote (VA), check remaining factors of the denominator:

    • Since the remaining denominator is (x+1)(x + 1),

    • Set it to zero: x+1=0⇒x=−1x + 1 = 0 \Rightarrow x = -1

Final Information

  • Vertical asymptote: x=−1x = -1

  • No oblique asymptotes since the function remains rational after cancellations.

Example 2: Another Rational Function

  • Given the function: g(x)=x2−25x3+125g(x) = \frac{x^2 - 25}{x^3 + 125}

Factoring and Identifying Holes

  • Factor both parts:

    • Numerator: (x+5)(x−5)(x + 5)(x - 5)

    • Denominator: (x+5)(x2−5x+25)(x + 5)(x^2 - 5x + 25)

  • Canceled factor: (x+5)(x + 5).

  • x-coordinate of hole: x=−5x = -5

  • New function: g(x)=x−5x2−5x+25g(x) = \frac{x - 5}{x^2 - 5x + 25}

  • Plugging x = -5:

    • g(−5)=−5−5−25+25+25=−1025=−25g(-5) = \frac{-5 - 5}{-25 + 25 + 25} = \frac{-10}{25} = -\frac{2}{5}

  • Therefore, the hole is at (-5, -2/5).

Vertical Asymptote Analysis

  • Observing the denominator: (x2−5x+25)(x^2 - 5x + 25) must be checked for real roots to find asymptotes.

  • The discriminant (D=b2−4acD = b^2 - 4ac) indicates whether roots exist:

    • D=(−5)2−4(1)(25)=25−100=−75D = (-5)^2 - 4(1)(25) = 25 - 100 = -75

    • Since D < 0, there are no vertical asymptotes.

Summary of Key Points

  • x-intercept occurs when the top equals zero.

  • y-intercept occurs when replacing x with zero in the function.

  • Steps to find these intercepts involve simple algebraic manipulation.

Additional Practice Problems

  • Practice finding vertical asymptotes, holes, x-intercepts, and y-intercepts for the rational function: d(x)=12x2−17x+76x2+25x−9d(x) = \frac{12x^2 - 17x + 7}{6x^2 + 25x - 9}

  • Evaluate the function to find intercepts and further derive holes or asymptotes, encouraging algebraic skills and rational function insights.