Chapter 1: Functions and Graphs

Section 1.1 & 1.2: Relations and Functions

Understanding relations and functions is foundational in precalculus. A relation pairs elements from one set with elements from another, while a function is a special relation where each input has exactly one output.

• Domain and Range: The domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• Evaluating Functions: Substitute the input value into the function to find the output.

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Section 1.3: More with Functions

Functions can be classified by their behavior and symmetry.

• Increasing, Decreasing, Constant: A function is increasing where its output rises as input increases, decreasing where output falls, and constant where output remains unchanged.

• Even and Odd Functions: An even function satisfies and is symmetric about the y-axis. An odd function satisfies and is symmetric about the origin.

• Piecewise Functions: Defined by different expressions for different intervals of the domain. Evaluate by determining which interval the input belongs to.

Section 1.4 & 1.5: Linear Functions

Linear functions describe straight lines and are fundamental in algebra and precalculus.

• Average Rate of Change: Measures how a function changes between two points:

• Difference Quotient: Used to analyze the rate of change:

• Forms of Linear Equations:

  • Slope-intercept form:

  • Standard form:

  • Point-slope form:

• Slope:

Section 1.6: Transformations of Functions

Transformations shift, stretch, or reflect the graph of a function. Common transformations include translations, reflections, and dilations.

• Translation: Shifts the graph horizontally or vertically.

• Reflection: Flips the graph over the x-axis or y-axis.

• Dilation: Stretches or compresses the graph.

Section 1.7: Combining and Composing Functions

Functions can be combined through arithmetic operations or composed to form new functions.

• Arithmetic: Addition, subtraction, multiplication, and division of functions.

• Composition: means applying first, then .

• Decomposition: Breaking a function into simpler functions.

Section 1.8: Inverse Functions

An inverse function reverses the effect of the original function. Not all functions have inverses.

• Finding the Inverse: Swap x and y, then solve for y.

• Horizontal Line Test: A function has an inverse if every horizontal line intersects its graph at most once.

Section 1.9: Circles

The equation of a circle describes all points equidistant from a center.

• Standard Form:

• General Form:

• Center: ; Radius:

Chapter 2: Polynomial and Rational Functions

Section 2.1: Complex Numbers

Complex numbers extend real numbers to include solutions to equations like .

• Standard Form: , where

• Operations: Addition, subtraction, multiplication, and division follow algebraic rules.

• Simplifying:

Section 2.2: Quadratic Functions

Quadratic functions are polynomials of degree 2 and have parabolic graphs.

• Standard Form:

• Vertex Form:

• Vertex Formula:

• Quadratic Formula:

• Solving: Factoring, quadratic formula, or taking square roots.

• Graphing: Identify vertex, axis of symmetry, direction of opening, and intercepts.

• Domain: All real numbers; Range: Depends on the direction of opening.

Section 2.3: Polynomial Functions

Polynomial functions are sums of powers of x with real coefficients.

• General Form:

• End Behavior: Determined by the leading term .

• Zeros: Values of x where ; can be found algebraically or graphically.

Section 2.4: Polynomial Functions Continued

Advanced techniques for analyzing polynomials include division and the Remainder Theorem.

• Long Division: Used to divide polynomials.

• Remainder Theorem: is the remainder when is divided by .

Section 2.5: Zeros of Polynomial Functions

Finding all zeros is essential for factoring and graphing polynomials.

• Factored Form: Express as a product of linear and irreducible quadratic factors.

• Standard Form:

Section 2.6: Rational Functions

Rational functions are quotients of polynomials and exhibit unique behaviors such as asymptotes and holes.

• Domain: All real numbers except where the denominator is zero.

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

• Holes: Occur where both numerator and denominator are zero for the same x-value.

• Horizontal/Slant Asymptotes: Determined by the degrees of numerator and denominator.

• Intercepts: x-intercepts where numerator is zero; y-intercept at .

• Symmetry: Even, odd, or neither, based on function properties.

Key Formulas and Theorems

• Distance Formula:

• Midpoint Formula:

• Pythagorean Theorem:

Example: Polynomial Function

Given , identify degree, leading coefficient, and end behavior.

• Degree: 3 (cubic)

• Leading Coefficient: 2

• End Behavior: As , ; as ,

Additional info: The notes above expand on brief formula sheets and study guide points, providing context and explanations for each topic. The included image directly illustrates the definition and general form of a polynomial function, matching the explanation in Section 2.3.