Quadratic Functions

Definition and General Form

Quadratic functions are polynomial functions of degree 2, typically written as:

• General Form: , where

Key Features of Quadratic Functions

• Vertex: The point where the parabola changes direction. Found using and .

• Axis of Symmetry: The vertical line .

• Y-intercept: The value .

• Direction of Opening: If , the parabola opens upward; if , it opens downward.

• Intervals of Increase/Decrease: For , decreasing on , increasing on ; for , increasing on , decreasing on .

Example

• Given :

  • , ,

  • Vertex: ;

  • Axis of symmetry:

  • Y-intercept:

  • Opens downward ()

Polynomial Functions

Definition and General Form

Polynomial functions are expressions of the form:

• Degree: The highest power of with a nonzero coefficient.

• Leading Coefficient: The coefficient of the highest degree term.

Key Concepts

• End Behavior: Determined by the degree and leading coefficient.

• Zeros: Values of where ; may be real or complex.

• Multiplicity: The number of times a zero is repeated.

• Descartes' Rule of Signs: Used to predict the number of positive, negative, and complex roots by counting sign changes in and .

Example Table: End Behavior

Example

• Given :

  • Degree: 4 (even)

  • Leading coefficient: 1 (positive)

  • End behavior: Both ends up

  • Use Descartes' Rule of Signs to analyze roots

Rational Functions

Definition and General Form

Rational functions are quotients of polynomials:

• , where

Key Features

• Vertical Asymptotes: Values of where and

• Horizontal/Oblique Asymptotes: Determined by the degrees of and

• Intercepts: Y-intercept at ; x-intercepts where

• Intervals of Positivity/Negativity: Solve and

Example

• Given :

  • Vertical asymptote:

  • Horizontal asymptote: Degree numerator = 2, denominator = 1; oblique asymptote exists

  • Y-intercept:

Graphing and Transformations

Parent Functions and Transformations

• Parent Functions: , , ,

• Transformations: Shifts, stretches, compressions, and reflections

Example

• Given :

  • Shift right by 1, up by 7, reflect over x-axis, vertical stretch by 3

Summary Table: Key Properties of Functions

Additional info: These notes expand on the provided questions by including definitions, formulas, and examples for each function type, as well as tables for classification and end behavior.