Polynomial and Rational Functions Notes

Polynomial Functions

• Section context: Chapter 1 Functions and Graphs; Section 4 Polynomial and Rational Functions

• Topics include: properties of polynomial functions, regression using a calculator, properties of rational functions, and applications of polynomial and rational functions.

• Definition: Polynomial Function

  • Domain: the set of all real numbers. ext{Domain}(p) = \, \mathbb{R}.
  • Range depends on degree:
  • If the degree is odd, the range is all real numbers. ext{Range}(p) = \mathbb{R}.
  • If the degree is even, the range is a proper subset of the real numbers.
  • Degree-based examples:
  • Constant function: polynomial of degree 0.
  • Linear function: polynomial of degree 1.
  • Quadratic function: polynomial of degree 2.

• Shared properties of odd degree polynomial graphs

  • All odd degree polynomials cross the x-axis at least once.
  • End behavior depends on the leading coefficient (the sign):
  • If the leading coefficient is positive, as x → −∞ the function tends to −∞ and as x → ∞ it tends to ∞.
  • If the leading coefficient is negative, as x → −∞ the function tends to ∞ and as x → ∞ it tends to −∞.

• Shared properties of even degree polynomial graphs

  • Some even polynomials never intersect the x-axis.
  • If the polynomial starts positive and has a positive leading coefficient, it ends positive.
  • If the polynomial starts negative and has a negative leading coefficient, it ends negative.
  • End behavior for even degree polynomials is the same on both ends (both go to +∞ or both to −∞, depending on the sign of the leading coefficient).

• Examples (graphing polynomials by degree)

  • Example 1: Graph a Polynomial of Degree 1.
  • Example 2: Graph a Polynomial of Degree 3.
  • Example 3: Graph a Polynomial of Degree 5.
  • Example 4: Graph a Polynomial of Degree 2.
  • Example 5: Graph a Polynomial of Degree 4.
  • Example 6: Graph a Polynomial of Degree 6.

• Observations about the graphs

  • Each odd-degree polynomial graph starts negative and ends positive when the leading coefficient is positive; ends move in opposite directions. It crosses the x-axis at least once.
  • Each even-degree polynomial graph starts positive and ends positive when the leading coefficient is positive; some cross the x-axis, some do not. Ends move in the same direction (up on both sides).

• Additional observations

  • No graph crosses the x-axis more times than its degree.
  • Changes in direction (local extrema) are limited to at most one less than the degree.

• Polynomials are continuous functions

  • Graphs of polynomials are continuous and smooth.
  • You can sketch all points on the graph without lifting the pen from paper.
  • The graph has no sharp corners (it is smooth).
  • Visual contrast: examples illustrating different levels of continuity: not continuous; continuous but not smooth; continuous and smooth.

• Regression polynomials

  • Section 1-3 covers quadratic regression as a regression polynomial model for data.
  • Many graphing calculators include other polynomial regression types as best-fit models for data sets.

• Example: Cubic regression (data modeling case)

  • Data context: marriage and divorce rates per 1000 people since 1960.
  • Let x represent the number of years since 1960. Task: create a scatter plot and find a cubic regression model y = f(x) that models the marriage rate.

• Creating a scatter plot (calculator workflow)

  • Input data into the statistics editor (Stat Edit).
  • L1 contains the number of years after 1960; L2 contains the marriage rate.
  • Set up a scatter plot: choose the type of plot and the data storage location in the calculator.
  • Delete any equations in the y = screen before viewing the scatter plot.
  • Use ZoomStat to size the graph window appropriately.
  • Resulting scatter plot is shown.

• Cubic regression modeling (calculator steps)

  • Access: Stat Calc → CubicReg (option 6).
  • Store the resulting model in Y1 (use: vars → Y-vars → Function → 1: Y1).
  • The model produced in the example:
  • Model: y = 8.70x^{3} - 0.01x^{2} + 0.29x + 8.55.
  • Goodness of fit: R^{2} = 0.995.

• Scatter plot vs cubic regression model

  • The cubic regression function closely matches the scatter plot data, indicating a good fit in this example.

Rational Functions

• Definition: Rational functions are quotients of polynomial functions.

  • If n(x) is a polynomial in the numerator and d(x) is a polynomial in the denominator, then the rational function is f(x) = rac{n(x)}{d(x)}.

• Example: Basic rational function domain

  • Let n(x) = x - 3 and d(x) = x - 2.
  • Since d(2) = 0, x = 2 is not in the domain of the rational function.
  • Domain: all real numbers except x = 2.

• Vertical asymptotes of rational functions

  • A vertical asymptote is a line x = h that the graph approaches but does not cross.
  • A vertical asymptote occurs when the denominator is zero (after reducing any common factors with the numerator).
  • Example statements:
  • The function has a vertical asymptote at x = 2.
  • The function has vertical asymptotes at x = -2 and x = 2.
  • The function has no vertical asymptotes.

• Horizontal asymptotes of rational functions

  • A horizontal asymptote is a line y = k that the graph approaches as x \to \pm \infty.
  • Example statements:
  • There is a horizontal asymptote at y = 1.
  • There is a horizontal asymptote at y = 0.
  • There is no horizontal asymptote.
  • Formal definition: as x\to \pm\infty, f(x) \to k for some constant k.

• Number of vertical asymptotes

  • If the denominator d(x) has degree n, the rational function can have no more than n vertical asymptotes.
  • If the numerator n(x) and denominator d(x) have no common real zeros, and if d(c) = 0, then x = c is a vertical asymptote.
  • If the numerator and denominator share common real zeros, these factors cancel; the remaining zeros of the denominator in the reduced form become the vertical asymptotes.

• Horizontal asymptotes (summary by degree comparison)

  • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.
  • If the degrees are equal, the horizontal asymptote is y = \frac{a}{b} where a and b are the leading coefficients of the numerator and denominator, respectively.
  • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

• Example: Find the asymptotes for a given rational function

  • Given:
  • n(x) = 3(x^{2} + x - 2) = 3(x - 1)(x + 2)
  • d(x) = 2(x^{2} - 1) = 2(x - 1)(x + 1)
  • Reduction: f(x) = \frac{n(x)}{d(x)} = \frac{3(x - 1)(x + 2)}{2(x - 1)(x + 1)} = \frac{3(x + 2)}{2(x + 1)}, \quad x \neq 1.
  • Vertical asymptote: The reduced denominator has a zero at x = -1. No other zeros remain, so x = -1 is the vertical asymptote.
  • Horizontal asymptote: The reduced numerator and denominator have the same degree and leading coefficients 3 and 2, respectively. Thus, the horizontal asymptote is y = \frac{3}{2}.
  • (Note: the original expression had a common factor x - 1 which cancels, indicating a hole at x = 1 in the unreduced form. This detail is implicit in the cancellation discussion.)

• Bounded functions

  • A function is bounded if its entire graph lies between two horizontal lines.
  • The only bounded polynomials are constant functions.
  • Many rational functions are bounded.

• Applications of rational functions

  • Example: A company manufacturing computers uses a model where a new employee can assemble N(t) components per day after t days of on-the-job training. The task is to sketch a graph of N for 0 \le t \le 100 and determine what N(t) approaches as t\to \infty.
  • Domain note: The domain may exclude points where the denominator is zero, depending on the model.
  • In the given example, the horizontal asymptote is y = 50, indicating a maximum rate approached over time.

• Solution to the example:

  • Vertical asymptotes: With domain t \ge 0, the function does not have vertical asymptotes in this domain because the denominator does not vanish for permissible t.
  • Horizontal asymptote: As t \to \infty, N(t) \to 50. Therefore, the horizontal asymptote is y = 50.

• Graph interpretation (Example): The graph of the rational function tends toward the horizontal asymptote y = 50 as time grows, suggesting the daily assembly rate approaches 50 components per day.

• Connections to earlier sections

  • Polynomial and rational functions extend from understanding basic functions and their graphs, domain/range, and end behavior.
  • Regression concepts provide practical modeling for data, including polynomial models for trends and rational functions for certain asymptotic behaviors.

• Ethical/philosophical/practical notes

  • Use of regression models requires awareness of data quality, overfitting risk with higher-degree polynomials, and the limits of extrapolation beyond observed data ranges.
  • When interpreting asymptotes, consider domain restrictions and the real-world meaning of limits in applied contexts (e.g., production rates).