Polynomial and Rational Functions - Study Notes

Polynomial Functions

• Definition: Polynomial Function
  • Domain: the set of all real numbers
  • Range:
  • For odd degree polynomials: all real numbers
  • For even degree polynomials: a proper subset of the real numbers
  • Special cases by degree:
  • Constant function — polynomial of degree 0
  • Linear function — polynomial of degree 1
  • Quadratic function — polynomial of degree 2

Graphical Properties of Polynomials

• Odd degree polynomial graphs (Shared properties):

  • All odd degree polynomials cross the x-axis at least one time
  • For odd degree with positive leading coefficient: start negative and end positive
  • For odd degree with negative leading coefficient: start positive and end negative

• Even degree polynomial graphs (Shared properties):

  • Some even polynomials never intersect the x-axis
  • If leading coefficient is positive and the graph starts positive, ends positive
  • If leading coefficient is negative and the graph starts negative, ends negative

Observations About Polynomial Graphs

• Examples graphing polynomials of various degrees (1, 3, 5, 2, 4, 6) illustrate end behavior and crossings
• Key observations:
  • Each odd degree polynomial graph starts negative and ends positive (assuming positive leading coefficient) and ends move in opposite directions
  • Each odd degree polynomial crosses the x-axis at least once
  • Each even degree polynomial graph starts positive and ends positive (assuming positive leading coefficient); some cross the x-axis, some do not; ends move in the same direction (up on both sides)
  • No graph crosses the x-axis more times than its degree
  • Changes in direction are limited to one less than the degree

Continuity and Smoothness

• Polynomials are Continuous Functions:
  • Graphs of polynomials are continuous and smooth
  • One can sketch all points on the graph without lifting the pen from paper
  • The graph has no sharp corners
• Related examples (non-polynomial graph types):
  • Not continuous graphs imply breaks in the curve
  • Continuous but not smooth graphs have cusps or corners
  • Continuous and smooth graphs have no breaks or corners

Regression Polynomials

• Regression polynomials overview:

  • Section 1-3 outlines the process of quadratic regression
  • Most graphing calculators include other types of polynomial regression as best-fit models for data sets

• Example: Cubic Regression (data context)

  • Data: marriage and divorce rates per 1,000 population for selected years since 1960
  • Let x represent the number of years since 1960
  • Task: Create a scatter plot and find a cubic regression model for the marriage rate

• Scatter plot creation (calculator steps):

  • Input the data into the statistics editor (Stat Edit)
  • L1 stores the number of years after 1960
  • L2 stores the marriage rate

• Setting up the scatter plot:

  • Use the stat plot menu to choose the type of plot and data storage locations

• Graphing the scatter plot:

  • If there are any existing equations in the y = screen, delete them before viewing the scatter plot
  • Use Zoom 9: ZoomStat to size the window and show the scatter plot
  • Resulting scatter plot is shown

• Creating a cubic regression model:

  • Access CubicReg under the Stat Calc menu (option 6)
  • Use appropriate settings in the calculator window
  • Store the regression model in Y1 via: ext{vars}
    ightarrow ext{Y-vars}
    ightarrow ext{Function}
    ightarrow 1:Y1

• Regression results (example):

  • Best-fit cubic regression model: y = 8.70x^{3} - 0.01x^{2} + 0.29x + 8.55
  • Coefficient of determination: R^{2} = 0.995 (approx. 99.5% fit)

• Interpretation:

  • The cubic regression function closely matches the scatter plot data
  • A high $R^{2}$ indicates a good fit for the model

Rational Functions

• Definition: Rational functions are quotients of polynomial functions

  • General form: f(x) = \frac{n(x)}{d(x)} where n(x) and d(x) are polynomials

• Example: Domain and Basic Domain Considerations

  • 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 f(x)
  • Domain: all real numbers except x = 2

• Vertical Asymptotes

  • A vertical asymptote of a rational function is a line of the form x = h that the graph approaches but does not cross
  • Example possibilities in the slides: a vertical asymptote at x = 2; at x = -2 and x = 2; or none

• Horizontal Asymptotes

  • A horizontal asymptote is a line y = k that the graph approaches as x\to \pm\infty
  • Examples: horizontal asymptotes at y = 1, y = 0, or none
  • General rule: for large |x|, the function approaches a constant if degrees satisfy certain relations
  • Definition: A horizontal asymptote of a rational function is a line of the form y = k which the graph approaches but does not cross as both x increases and decreases without bound

• Number of Vertical Asymptotes

  • If the denominator degree is n, a rational function can have at most n vertical asymptotes
  • If the numerator and denominator have no common real zeros and d(c) = 0, then x = c is a vertical asymptote
  • If the numerator and denominator share common real zeros, these common factors can be cancelled; any remaining real zeros of the reduced denominator are vertical asymptotes

• Horizontal Asymptotes (in detail)

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

• Example: Find Asymptotes (factorization)

  • Given:
  • n(x) = 3(x^{2} + x - 2) = 3(x - 1)(x + 2)
  • d(x) = 2(x^{2} - 1) = 2(x - 1)(x + 1)
  • The function reduces by cancelling the common factor x - 1 to
  • Reduced form: f(x) = \frac{3(x + 2)}{2(x + 1)}
  • Vertical asymptote: denominator of the reduced function has a zero at x = -1; hence the vertical asymptote is x = -1
  • Horizontal asymptote: degrees of numerator and denominator are both 2; leading coefficients are 3 and 2; hence y = \dfrac{3}{2}

• 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: Example with N(t) for production (rational function context)

  • Scenario: A company that manufactures computers has established that a new employee can assemble N(t) components per day after t days of on-the-job training, where the domain is 0 \le t \le 100
  • Question: Sketch a graph of N(t) and determine what N(t) approaches as t increases without bound

• Solution (Asymptotes for this application)

  • Vertical asymptote: With domain t \ge 0, there is no vertical asymptote for the function given the stated domain
  • Horizontal asymptote: For large t, N(t) approaches the leading coefficient ratio of the numerator to the denominator, here y = 50
  • In general, if numerator leading coefficient is a and denominator leading coefficient is b for large t, the horizontal asymptote is y = \dfrac{a}{b}

• Graph of the application

  • As t increases, N(t)\to 50 components per day; the value 50 represents the upper limit expected for daily output

Summary of Key Formulas and Concepts

• Polynomial Function basics:

  • Domain: all real numbers
  • Range depends on degree: odd → all real; even → subset of real numbers
  • Degrees correspond to the function’s shape and end behavior

• End behavior by degree:

  • Odd degree: as x\to -\infty and x\to \infty, the ends go in opposite directions; sign depends on leading coefficient
  • Even degree: ends go in the same direction; sign depends on leading coefficient

• Continuity and smoothness:

  • Polynomials are continuous for all real numbers and are smooth (no sharp corners)

• Regression polynomials:

  • Quadratic regression is common; calculators can perform higher-degree polynomial regression (including cubic)
  • Goodness of fit measured by R^{2}, where higher values indicate a better fit

• Rational functions:

  • Domain excludes zeros of the denominator
  • Vertical asymptotes occur where the function goes to infinity; can be eliminated by cancelling common factors if they are present
  • Horizontal asymptotes are determined by degree comparison between numerator and denominator
  • Boundedness: polynomials that are constant are bounded; many rational functions are bounded

• Applications:

  • Use asymptotes to understand long-term behavior of real-world models (e.g., production rates approaching a limit)