Functions and Their Graphs

Vocabulary and Concepts

Understanding the basic terminology and concepts related to functions is essential in precalculus. Functions describe relationships between variables, typically written as .

• Function: A relation in which each input (domain) has exactly one output (range).

• Graph of a Function: A visual representation of all ordered pairs .

• Domain and Range: The set of all possible input values (domain) and output values (range).

• Example: For , the domain is all real numbers, and the range is .

Properties of Functions

Functions have various properties that help describe their behavior and characteristics.

• Increasing/Decreasing: A function is increasing if for ; decreasing if .

• Even/Odd Functions: Even functions satisfy ; odd functions satisfy .

• One-to-One: A function is one-to-one if each output is paired with only one input.

• Example: is odd and one-to-one; is even but not one-to-one.

Transformations of Functions and Graphs

Types of Transformations

Transformations alter the appearance of a function's graph.

• Translations: Shifting the graph up/down or left/right.

• Reflections: Flipping the graph over the x-axis or y-axis.

• Stretching/Compressing: Changing the graph's width or height.

• Example: is shifted right by 2 and up by 3.

Graphing Using Transformations

Given a function definition, transformations can be applied to graph the function efficiently.

• Step 1: Identify the base function (e.g., ).

• Step 2: Apply horizontal and vertical shifts, reflections, and stretches/compressions as indicated.

• Example: Graph by reflecting, stretching, and shifting.

Linear Functions and Models

Identifying Linearity

Linear functions have constant rates of change and are represented by straight lines.

• General Form:

• Linearity from Table: If the difference in values is constant for equal changes in , the function is linear.

• Example: Table: ; (difference is 2, so linear).

Applications of Linear Models

Linear models are used to describe real-world relationships with constant rates of change.

• Example: Cost as a function of number of items: .

Quadratic Functions

Graphing Quadratics

Quadratic functions have parabolic graphs and can be analyzed using their vertex, axis of symmetry, and intercepts.

• Standard Form:

• Vertex:

• Axis of Symmetry: Vertical line through the vertex.

• Intercepts: Set for -intercept; set for -intercepts.

• Example: has vertex at .

Finding Quadratic Equations

Given points, vertex, or intercepts, you can construct the equation of a quadratic function.

• Vertex Form: where is the vertex.

• Example: Vertex , passes through : ; solve for using the point.

Polynomial Functions

Identifying Graphs and Zeros

Polynomial functions can be classified by degree and analyzed for real zeros.

• Degree: Highest power of in the polynomial.

• Real Zeros: Values of where .

• Example: has degree 3 and zeros at (additional info: solve for all zeros).

Transformations of Polynomial Functions

Transformations can be applied to polynomial functions similarly to other functions.

• Example: is shifted right by 2 and up by 1.

Intercepts, Asymptotes, and Zeros of Rational Functions

Rational functions are quotients of polynomials and have unique features such as asymptotes.

• Intercepts: Set for -intercept; set numerator for -intercepts.

• Vertical Asymptotes: Set denominator and solve for .

• Horizontal Asymptotes: Compare degrees of numerator and denominator.

• Example: has vertical asymptote at .

Graphing and Solving Polynomial and Rational Inequalities

Solving inequalities involves finding intervals where the function is positive or negative.

• Example: Solve ; solution: or .

Theorems for Complex Zeros

Polynomial functions may have complex zeros, especially for higher degrees.

• Fundamental Theorem of Algebra: Every polynomial of degree has complex zeros (counting multiplicities).

• Example: has zeros at and .

Evaluating Functions and Domains

Evaluating Functions

To evaluate a function, substitute the given value for .

• Example: , find : .

Domain of Rational Functions

The domain of a rational function excludes values that make the denominator zero.

• Example: ; domain is all real numbers except .

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form , where and are constants.

• Growth/Decay: If , the function grows; if , it decays.

• Example: is exponential growth.

Finding Equations from Graphs

Given points or a graph, you can determine the equation of an exponential function.

• Example: If and , solve and to find and .

Applications of Exponential and Logarithmic Functions

These functions model growth, decay, and other real-world phenomena.

• Example: Population growth:

• Logarithmic Functions: Inverse of exponential functions,

Summary Table: Key Features of Function Types

Additional info: Academic context and examples have been added to expand on the brief points in the original study guide, ensuring completeness and clarity for exam preparation.