Polynomial Functions

Identifying Polynomial Graphs

Polynomial functions are algebraic expressions involving powers of x with real coefficients. The graph of a polynomial function is a smooth, continuous curve.

• Key features: Degree, leading coefficient, zeros (roots), and end behavior.

• Example: Given a graph, match it to its equation by analyzing the number and location of zeros, turning points, and end behavior.

Additional info: The degree of a polynomial determines the maximum number of turning points (degree - 1).

Properties of Polynomial Functions

Understanding the leading coefficient and degree helps predict the graph's behavior.

• Leading coefficient: Determines the direction of the ends of the graph.

• Degree: Even degree polynomials have ends going in the same direction; odd degree polynomials have ends going in opposite directions.

• Example: If a graph has both ends down, the degree is even and the leading coefficient is negative.

Rational Functions

Horizontal Asymptotes

Rational functions are quotients of polynomials. The horizontal asymptote describes the end behavior as x approaches infinity.

• Rule:

  • If degree of numerator < degree of denominator: asymptote at

  • If degrees are equal: asymptote at

  • If degree of numerator > degree of denominator: no horizontal asymptote

• Example: has a horizontal asymptote at .

Identifying Rational Functions from Graphs

Graphs of rational functions often have vertical and horizontal asymptotes, and may have holes or jumps.

• Vertical asymptotes: Occur where the denominator is zero and the numerator is not zero.

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

• Example: Given a graph with vertical asymptotes at and , and a horizontal asymptote at , the function could be .

Asymptotes and Holes

For a rational function , vertical asymptotes occur at zeros of not canceled by . Holes occur where both and have a common factor.

• Example:

• Vertical asymptotes: Solve

• Horizontal asymptote: (if degrees are equal)

Exponential and Logarithmic Functions

Evaluating Logarithms

Logarithms are the inverse of exponentials. The logarithm answers the question: "To what power must be raised to get ?"

• Key properties:

• Example: because .

Expanding Logarithmic Expressions

Use logarithm properties to expand or simplify expressions.

• Product rule:

• Quotient rule:

• Power rule:

• Example:

Solving Exponential and Logarithmic Equations

To solve equations involving exponentials or logarithms, use properties of logarithms and exponentials to isolate the variable.

• Example: Solve by equating exponents:

• Example: Solve by combining logs:

Applications: Compound Interest and Exponential Growth

Compound Interest

Compound interest is calculated using the formula:

• P: Principal (initial amount)

• r: Annual interest rate (decimal)

• n: Number of times interest is compounded per year

• t: Number of years

• Example: Find the amount after 1 year for $400 compounded daily:

Continuous Compound Interest

When interest is compounded continuously, use:

• Example: for a $5000 for $18$ years.

Exponential Growth and Decay

Exponential growth is modeled by , where is the growth rate.

• Example: An ant colony starts with $800 per day:

• Decay: Half-life problems use , where

• Example: If the half-life of radium is $1600k = \frac{\ln 2}{1600}$

Summary Table: Key Properties of Polynomial and Rational Functions