Exponents and Their Properties

Definition and Laws of Exponents

Exponents are a way to represent repeated multiplication of a base number. Understanding the rules of exponents is essential for simplifying expressions and solving equations in precalculus.

• Product of Powers:

• Quotient of Powers: , where

• Power of a Power:

• Power of a Product:

• Power of a Quotient: , where

• Zero Exponent: , where

• Negative Exponent: , where

Example: Simplify .

• Apply the quotient rule:

Roots and Rational Exponents

Definition and Properties

Roots are the inverse operation of exponents. Rational exponents provide a way to express roots as exponents.

• n-th Root:

• Rational Exponents:

• Product and Quotient Rules: ,

Example: Simplify .

• Prime factorization:

• Pair the factors:

Solving Quadratic Equations

Factoring and Special Products

Quadratic equations are equations of the form . They can be solved by factoring, using the quadratic formula, or by completing the square.

• Difference of Squares:

• Perfect Square Trinomial:

• Zero Product Property: If , then or

Steps to Solve by Factoring:

1. Set the equation to zero:

2. Factor the quadratic expression

3. Set each factor equal to zero and solve for

Example: Solve .

• Factor:

• Set each factor to zero: ,

Functions and Their Properties

Definition and Notation

A function is a relation in which each input (x-value) has exactly one output (y-value). Function notation is written as , where is the input.

• Function: Each -value corresponds to only one -value.

• Not a Function: An -value corresponds to more than one -value.

• Function Notation: denotes the output when is the input.

Domain and Range

The domain of a function is the set of all possible input values (-values). The range is the set of all possible output values (-values).

• To find the domain:

  • Start with all real numbers

  • Exclude values that make the denominator zero

  • Exclude values that make the expression under an even root negative

• Example:

  • Denominator is excluded

  • Domain:

Evaluating Functions

To evaluate a function, substitute the given value for in the function's formula.

• Example: If , then

Graphs of Functions

Key Features of Graphs

Understanding the graphical representation of functions is crucial for analyzing their behavior.

• Domain: The set of -values for which the function is defined

• Range: The set of -values the function attains

• Intercepts:

  • x-intercept: Where

  • y-intercept: Where

• Symmetry:

  • Even function: (symmetric about the y-axis)

  • Odd function: (symmetric about the origin)

  • Neither: No symmetry

• Intervals of Increase/Decrease/Constant:

  • Increasing: -values rise as increases

  • Decreasing: -values fall as increases

  • Constant: -values remain the same as increases

• Local Maximum/Minimum: Highest/lowest points in a local region (not at endpoints)

Example: For a function with domain and range , the graph starts at and ends at , with -values between just above 2 and 4.

Library of Functions and Piecewise Functions

Common Functions

• Linear:

• Quadratic:

• Cubic:

• Absolute Value:

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals of the domain.

• To evaluate, determine which interval the input belongs to and use the corresponding formula.

• Example:

• Find : Use

• Find : Use

Transformations of Functions

Types of Transformations

Transformations change the position or shape of a function's graph.

• Vertical Shifts: shifts up by ; shifts down by

• Horizontal Shifts: shifts left by ; shifts right by

• Vertical Stretch/Compression: stretches if , compresses if

• Reflections: reflects across the x-axis; reflects across the y-axis

Steps for Graphing Transformations:

1. Identify the base function

2. Determine the sequence of transformations

3. Apply transformations to the base graph

4. Plot the new graph

Example:

• Base:

• Shift left by 7 units, up by 4 units

Summary Table: Key Function Properties

Additional info: Some steps and explanations were expanded for clarity and completeness, including the summary table and more detailed examples of function evaluation and transformations.