Functions and Their Properties

Polynomial Functions

Polynomial functions are algebraic expressions consisting of variables and coefficients, involving only non-negative integer powers of the variable.

• Definition: A polynomial function has the form , where is a non-negative integer and coefficients are real numbers.

• Key Properties:

  • Exponents must be whole numbers (no fractions or negatives).

  • No variables in denominators or under roots.

• Example: is a polynomial; is not.

Rational Functions and Asymptotes

Rational functions are quotients of polynomials. Their graphs may have vertical and horizontal asymptotes.

• Vertical Asymptote: Occurs where the denominator is zero and the numerator is nonzero. Example: For , the vertical asymptote is .

• Horizontal Asymptote: Determined by the degrees of numerator and denominator.

  • If degree numerator < degree denominator:

  • If degrees are equal: (leading coefficient numerator)/(leading coefficient denominator)

  Example: For , degree numerator < degree denominator, so .

Function Transformations and Reflections

Transformations include shifts, stretches, compressions, and reflections.

• Reflection over x-axis: Multiply the function by -1.

• Reflection over y-axis: Replace with .

• Example: is a reflection over the x-axis of .

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form , where is a constant and is the base.

• Parent Function:

• Transformations: shifts the graph right by and up by .

• Domain and Range:

  • Domain:

  • Range: for

• Horizontal Asymptote:

• Example: has horizontal asymptote .

Evaluating Exponential Expressions

• Example: (rounded to nearest hundredth)

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions.

• Definition: , where ,

• Domain: for ; for , domain is .

• Inverse Function: If , then

Solving Logarithmic Equations

• Change of Form:

• Example:

Condensing Logarithmic Expressions

• Properties:

• Example:

Trigonometry

Angles and Radian Measure

• Coterminal Angles: Angles that share the same terminal side. Example: and are coterminal because .

• Supplementary Angles: Two angles whose sum is . Example: Supplement of is .

• Arc Length: , where is radius and is angle in radians. Example: For ft, rad,

Unit Circle and Trigonometric Values

• Coordinates: On the unit circle, corresponds to .

• Example: , sine value is .

• Exact Values:

Special Triangles

• 30-60-90 Triangle:

  • Hypotenuse:

  • Short leg:

  • Long leg:

  Example: If hypotenuse is $14.

• Right Triangle:

  • Pythagorean Theorem:

  • Example: Legs $6, hypotenuse

Trigonometric Equations and Identities

• Solving Equations:

  • ,

• Period and Phase Shift:

  • Period of is

  • Example: , period is

  • Phase shift: , rewrite as

• Graph Behavior:

  • Tangent repeats every units, sine every units.

  • Cosecant approaches infinity near its asymptotes.

Summary Table: Key Properties of Functions

Additional info:

• Some questions involve graphical analysis and transformations, which are essential for understanding function behavior in precalculus.

• Problems cover all major precalculus topics: polynomials, rational functions, exponential and logarithmic functions, trigonometry, and function transformations.