Functions and Their Graphs

Understanding Functions

Functions are fundamental objects in mathematics that assign each input exactly one output. Their graphs visually represent the relationship between variables.

• Definition of a Function: A function f from set A to set B assigns each element of A exactly one element of B.

• Domain and Range: The domain is the set of all possible input values; the range is the set of all possible output values.

• Increasing/Decreasing Functions: A function is increasing on an interval if, as x increases, f(x) increases; decreasing if f(x) decreases.

• Intercepts: The points where the graph crosses the axes (x-intercept: f(x)=0; y-intercept: x=0).

• Symmetry: Functions may be even (symmetric about the y-axis) or odd (symmetric about the origin).

• Transformations: Shifting, stretching, compressing, and reflecting graphs.

• Piecewise Functions: Defined by different expressions over different intervals.

Example: The function is even and its graph is a parabola opening upwards.

Linear and Quadratic Functions

Linear Functions

Linear functions have the form and their graphs are straight lines.

• Slope: represents the rate of change; is the y-intercept.

• Applications: Modeling constant speed, cost, or growth.

Quadratic Functions

Quadratic functions have the form and their graphs are parabolas.

• Vertex: The highest or lowest point, at .

• Axis of Symmetry: The vertical line .

• Factoring and Completing the Square: Methods for solving quadratic equations.

Example: factors to .

Polynomial and Rational Functions

Polynomial Functions

Polynomials are sums of powers of x with coefficients. Their degree determines the number of roots and the end behavior.

• End Behavior: Determined by the leading term.

• Zeros: Solutions to .

• Graphing: Use intercepts and turning points.

Rational Functions

Rational functions are ratios of polynomials.

• Vertical Asymptotes: Where the denominator is zero.

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

Example: has a hole at .

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form .

• Growth and Decay: Used to model population, radioactive decay, and compound interest.

Logarithmic Functions

Logarithms are the inverses of exponentials: means .

• Properties:

• Applications: pH, Richter scale, sound intensity.

Trigonometric Functions

Basic Trigonometric Functions

Trigonometric functions relate angles to ratios of sides in right triangles and to points on the unit circle.

• Sine, Cosine, Tangent: , ,

• Unit Circle: All trigonometric values for common angles.

• Periodicity: Sine and cosine have period .

Example:

Analytic Trigonometry

Identities and Equations

Trigonometric identities are equations true for all values of the variable.

• Pythagorean Identity:

• Double Angle:

• Solving Equations: Use identities and algebraic manipulation.

Applications of Trigonometric Functions

Modeling and Problem Solving

Trigonometric functions model periodic phenomena such as sound, light, and tides.

• Law of Sines:

• Law of Cosines:

• Angular Speed:

Polar Coordinates and Vectors

Polar Coordinates

Polar coordinates represent points by their distance from the origin and angle from the positive x-axis.

• Conversion: ,

• Graphs: Circles, spirals, and other curves.

Vectors

Vectors have magnitude and direction, and are used to represent quantities such as force and velocity.

• Component Form:

• Magnitude:

Analytic Geometry

Conic Sections

Conic sections are curves formed by the intersection of a plane and a cone: circles, ellipses, parabolas, and hyperbolas.

• Standard Equations:

  • Circle:

  • Ellipse:

  • Parabola: or

  • Hyperbola:

• Vertices, Foci, Axes: Key features for graphing and analysis.

Table: Conic Section Equations and Properties

Systems of Equations and Inequalities

Solving Systems

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Methods: Substitution, elimination, and graphing.

• Applications: Intersection points, optimization problems.

Formula Sheet Highlights

Key Formulas Provided

• Unit Circle: Coordinates for common angles, values of sine and cosine.

• Conic Sections: Standard equations for circle, ellipse, parabola, hyperbola.

• Law of Sines and Cosines: For solving triangles.

• Arc Length:

• Angular Speed:

• Linear Speed:

Additional info:

• Some tables and diagrams were inferred to clarify conic section properties and polar coordinate conversions.

• All major Precalculus topics are covered, including functions, trigonometry, analytic geometry, and systems of equations.