Study Notes on Polynomial Functions of Higher Degree

Polynomial Functions of Higher Degree

Overview

• Subject Matter: Transformation skills to sketch polynomial graphs, Leading Coefficient Test for end behavior, finding and using zeros for sketches, applying Intermediate Value Theorem for locating zeros.

Learning Objectives

• Use transformations to sketch graphs of polynomial functions.

• Apply the Leading Coefficient Test to ascertain end behavior of polynomial graphs.

• Identify and utilize zeros of polynomial functions as aids in sketching.

• Implement the Intermediate Value Theorem to locate zeros of polynomial functions.

Graphs of Polynomial Functions

Matching Functions with Graphs

• Warm-up Exercise: Participants are tasked with matching polynomial functions with their respective graphs, explaining reasoning and verifying with graphing calculators.

  • a. f(x) = x³ - x

  • b. f(x) = -x³ + x

  • c. f(x) = -x + 1

  • d. f(x) = x⁴

  • e. f(x) = x³

  • f. f(x) = x⁴ = x²

Characteristics of Polynomial Functions

• End Behavior: The end behavior describes how the graph behaves as x approaches positive or negative infinity.

  • Example for f(x): Positive large values yield large outputs when substituting large positive inputs.

  • Function specifics include:

  • X-intercepts: -1, 0, 1.

  • Y-intercept: 1.

  • Behavior: Rises left to right and falls on both ends.

Graphing Polynomial Functions by Degree

• Functions can be represented as follows:

  • Degree 0: Constant function - Horizontal line.

  • Degree 1: Linear function - Line of slope a.

  • Degree 2: Quadratic function - Parabola.

  • Higher degree functions (degree > 2) have more complex graphs.

Properties of Graphs

• Continuity: Polynomial function graphs are continuous (no breaks, holes, or gaps).

  • Graphs exhibit smooth, rounded turns (sharp turns indicate non-polynomial functions).

Polynomial Function Formulas

• General form of polynomial function of degree n: $f(x) = a<em>n x^n + a</em>{n-1}x^{n-1} + ext{…} + a<em>2x^2 + a</em>1x + a_0$

  • Where n is a positive integer and $a_n <br>eq 0$.

End Behavior and Leading Coefficient Test

Understanding End Behavior

1. Odd Degree Functions:

   • If a_n > 0: Falls to the left, rises to the right.

   • If a_n < 0: Rises to the left, falls to the right.

2. Even Degree Functions:

   • If a_n > 0: Rises to the left, rises to the right.

   • If a_n < 0: Falls to the left, falls to the right.

Leading Coefficient Test

• The polynomial function’s degree and leading coefficient dictate its end behavior as described above, allowing predictions about the graph’s long-run behavior.

Example of Applying the Leading Coefficient Test

• For a polynomial $f(x) = -x^3 + 4x$ with odd degree and negative leading coefficient:

  • Result: Rises to the left and falls to the right.

Zeros of Polynomial Functions

Understanding Zeros

1. Zeros are defined as follows:

   • x = a is a zero of the polynomial function f.

   • It is also a solution to the equation $f(x) = 0$.

   • The expression $(x - a)$ is a factor of f.

   • (a, 0) is the corresponding x-intercept of the graph of f.

Finding Real Zeros Example

• To find zeros of $f(x) = x^3 - x^2 - 2x$:

  1. Rewrite the function: $0 = x^3 - x^2 - 2x$.

  2. Factor out common terms: $0 = x(x^2 - x - 2)$.

  3. Factor completely: $0 = x(x - 2)(x + 1)$.

Key Concepts of Multiplicity

• Multiplicity of Zeros:

  • Factor of (x − a)^k indicates a zero of multiplicity k.

  • If k is odd, the graph crosses the x-axis at x = a.

  • If k is even, the graph touches the x-axis and does not cross.

Applicability of Theorem in Real-world Problems

• Intermediate Value Theorem states that for any two points on the graph, if the function values differ, there exists a c where the function takes that value.

Finding Intervals with Zeros

• For the function $f(x) = 12x^3 - 32x^2 + 3x + 5$, find two points where the function changes signs, indicating a zero.

Conclusion

• Lastly, understanding polynomial functions involves thorough engagement with their graphs, the leading coefficients, their zeros, and the continuity derived from their definitions. The interplay between graphical characteristics and algebraic factors forms a fundamental aspect of polynomial function analysis.

Overview

• Subject Matter: Transformations to sketch polynomial graphs, Leading Coefficient Test for end behavior, finding zeros, applying Intermediate Value Theorem.

Learning Objectives

• Use transformations to sketch graphs.

• Apply Leading Coefficient Test for end behavior.

• Identify zeros of polynomial functions for sketches.

• Use the Intermediate Value Theorem to locate zeros.

Graphs of Polynomial Functions

• Matching Functions with Graphs: Match polynomial functions with graphs using reasoning and verification with calculators.

• Characteristics:

  • End Behavior: Positive outputs for large positive inputs, X-intercepts: -1, 0, 1; Y-intercept: 1; Rises left to right.

Properties

• Continuity: All polynomial graphs are continuous, with smooth turns.

Function Forms

• General form: $f(x) = a<em>n x^n + a</em>{n-1}x^{n-1} + … + a_0$.

End Behavior and Leading Coefficient Test

1. Odd Degree Functions:

   • a_n > 0: Falls left, rises right.

   • a_n < 0: Rises left, falls right.

Zeros of Polynomial Functions

1. Understanding Zeros:

   • Zeros are solutions to $f(x) = 0$, represented as factors of the polynomial.

Conclusion

• Understanding polynomials requires engagement with graphs, leading coefficients, and zeros, integrating graphical and algebraic aspects.