Writing Polynomial Function Formulas

Writing Formulas for Polynomial Functions

Learning Outcomes

  • Understanding how to write a polynomial function based on its graph and identifying its zeros.
  • Recognizing that a polynomial in factored form has x-intercepts at the zeros of its factors.

General Note: Factored Form of Polynomials

  • If a polynomial of the lowest degree p has zeros at x = x₁, x₂, …, xₙ, it can be expressed as:
    • f(x) = a(x − x₁)²(x − x₂)²… (x − xₙ)² where pᵢ indicates the multiplicity of each factor.
    • The stretch factor 'a' adjusts the polynomial vertically, determined by known points on the polynomial.

Steps to Write a Formula from a Graph

  1. Identify x-intercepts: Locate where the graph crosses the x-axis. These give initial factors for the polynomial.
  2. Examine graph behavior at intercepts: Determine if the polynomial bounces off (indicating even multiplicity) or passes through (indicating odd multiplicity).
  3. Construct polynomial of least degree: Combine identified factors based on their multiplicities.
  4. Locate stretch factor: Use another point from the graph, often the y-intercept, to solve for 'a'.

Example: Writing a Formula for a Polynomial Function

  • Suppose a graph has x-intercepts at x = –3, x = 2, and x = 5.
  • Y-intercept is at (0, -2).
  • X-intercepts behavior:
    • At x = -3 and x = 5, the graph crosses the axis linearly → linear factors.
    • At x = 2, it bounces → quadratic factor.
  • Polynomial resulting from these intercepts:
    • f(x) = a(x + 3)(x - 2)²(x - 5)
Finding 'a'
  • Plugging the y-intercept into the formula:
    • f(0) = a(0 + 3)(0 - 2)²(0 - 5) = -2
    • This simplifies to: -2 = a(3)(4)(-5) → -2 = -60a → a = 1/30
  • Final Function:
    • f(x) = (1/30)(x + 3)(x - 2)²(x - 5)

Local and Global Extrema

  • Extrema Definition:
    • Local Maximum: Highest point within a neighborhood.
    • Local Minimum: Lowest point within a neighborhood.
    • Global Maximum/Minimum: Highest/lowest point overall.
Conditions for Extrema
  • A polynomial function of even degree has a global maximum/minimum.
  • Example function f(x) = x (odd degree) has neither.

Applications of Local Extrema

  • Example Problem: Determine box volume from cutting squares from a plastic sheet.
  • Steps to Solve:
    1. Visualize the cut-outs with variable w.
    2. Calculate the volume function V(w) = (20 - 2w)(14 - 2w)(w)
    3. Identify realistic domain for w (should be between 0 and 7).
  • Max volume approximately occurs when squares are cut 2.75 cm, refining down to about 2.7 cm for more accurate volume being 339 cubic cm.

Exploring Leading Coefficients

  • Change the leading coefficient 'a' to see how it affects polynomial's shape (end behavior and intercepts).
  • Observe effects of both positive and negative values of 'a'.
  • Example outcomes: maximum occurs around (5.98, -398.8) and minimum around (0.02, 3.24).