Guided Notes for Polynomial Functions

Introduction to Polynomial Functions

  • Polynomial Definition: A polynomial function is an expression involving a sum of powers in one or more variables multiplied by coefficients.

Identifying Polynomial Functions

  1. Are the following polynomial functions? (Yes/No)
    • a. f(x) = -3x^4 + 2x^2 - 10
      • Yes
      • Degree: 4
      • Leading Coefficient (LC): -3
    • b. f(x) = x^2 + 7x^3 + x
      • Yes
      • Degree: 3
      • Leading Coefficient: 7
    • c. f(x) = 8√x + x - 5
      • No (contains a radical term)
    • d. f(x) = x^2(x + 2)
      • Yes
      • Degree: 3 (sum of exponents)
      • LC: 1
    • e. f(x) = -x^{-5}
      • No (negative exponent)
  2. Definitions:
    • Degree: Highest power of x.
    • Leading Coefficient: Number in front of the highest power of x.

End Behavior of Polynomial Functions

  • Definition: The end behavior of a polynomial function describes how the function behaves as x approaches positive or negative infinity.
  • Determining End Behavior:
    • If LC > 0 (positive):
    • Degree Even: Rises left and right.
    • Degree Odd: Falls left, rises right.
    • If LC < 0 (negative):
    • Degree Even: Falls left and right.
    • Degree Odd: Rises left, falls right.

Example: Determine End Behavior

  1. f(x) = x^5 - 5x + 4
    • Degree: 5 (odd)
    • LC: 1 (positive)
    • End Behavior: Falls to left, rises to right.
  2. g(x) = 3x^4 - 4x^2 - 4x + 1
    • Degree: 4 (even)
    • LC: 3 (positive)
    • End Behavior: Rises to left and right.

Finding Zeros of Polynomial Functions

  • Zeros: Solutions to the equation f(x) = 0, corresponding to x-intercepts.
  • Example:
  1. f(x) = 5(x - 6)(x + 3)^2(2x - 5)
    • Zeros: 6, -3, 5/2
    • Calculation: Set each factor to zero.
  2. g(x) = -3x(x^2 - 4)(x^2 + 36)
    • Zeros: 0, ±2
    • Note: x^2 + 36 has no real solutions.

Finding x- and y-intercepts

  • x-intercepts: Solve f(x) = 0.
  • y-intercept: Evaluate f(0).

Finding Zeros and Their Multiplicities

  • If a factor (x - c) appears k times, c is a zero of multiplicity k.
  • Multiplicity Implications:
    • Odd Multiplicity: The graph crosses the x-axis at c.
    • Even Multiplicity: The graph touches the x-axis at c.

Matching Graphs with Polynomial Functions

  1. Steps to Sketch a Polynomial:
    • Determine end behavior using degree and leading coefficient.
    • Identify zeros (x-intercepts) and their multiplicities.
    • Calculate y-intercept by evaluating f(0).
    • Draw a sketch based on left end behavior and connect intercepts accordingly (cross or touch the x-axis as per multiplicity).

Graphing Examples

  1. f(x) = (x - 1)(x + 3)(x - 2)
    • End Behavior: Degree = 4 (even), LC = 1 (positive)
    • Zeros: 1 (touch), -3 (cross), 2 (cross).
  2. f(x) = -2x^3 + 4x^2 + 6x
    • End Behavior: Degree = 3 (odd), LC = -2 (negative)
    • Zeros: Cross at 0, -3, and 1.