Pre-Calculus_Unit_2_Functions_Graphs

Chapter 1: Introduction to Precalculus

  • Welcome to precalculus and the exploration of functions and graphs.

  • Objective: Rediscover core concepts of precalculus and appreciate the beauty of algebra.

Relations and Functions

  • Relation: A set of ordered pairs connecting inputs and outputs. Can be represented in various forms (e.g., tables, mappings, graphs, equations).

  • Domain: Set of all possible inputs for a relation.

  • Range: Set of all possible outputs for a relation.

  • Example: Vending machine analogy - domain is buttons, range is snacks.

  • Function: Special relation where each input (domain) has exactly one output (range).

Chapter 2: Zeros of Functions

Checking Functions' Validity

  • Vertical Line Test: If a vertical line crosses a graph more than once, it is not a function.

  • Useful for analyzing and visualizing data relationships.

Understanding Function Notation

  • Function Notation: Represented as f(x), showing how function f transforms input x into output.

  • Example: For f(x) = 2x + 1, if x = 3, then f(3) = 7.

Characteristics of Functions

Intercepts

  • Intercepts: Points where the graph crosses the x-axis (x-intercepts) or y-axis (y-intercepts).

  • X-intercepts are also called zeros since at these points, f(x) = 0.

Zeros of a Function

  • Finding zeros requires solving the function equation set to 0 (algebraically) or identifying where the graph intersects the x-axis (graphically).

Extrema

  • Extrema: Maximum or minimum points on a function's graph, crucial for optimization problems (e.g., profit maximization).

Chapter 3: Finding Intercepts and Zeros

Intercept and Zero Discovery

  • Algebraic methods for finding intercepts and zeros involve setting equations to zero and evaluating functions.

  • Graphical methods involve visually identifying where graphs cross axes.

Understanding Extrema

  • Extrema represent peaks (maximum) and valleys (minimum) on graphs and are critical for optimization.

Increasing and Decreasing Intervals

  • Increasing Intervals: Where the function is rising as you move from left to right.

  • Decreasing Intervals: Where the function is falling.

  • Visualize function behavior like stock market trends.

Chapter 4: Function Characteristics

Continuity

  • Continuous Function: Graph has no breaks, jumps, or holes (drawn without lifting pencil).

  • Types of discontinuities: removable, jumped, and infinite.

End Behavior

  • End Behavior: Describes function behavior as x approaches positive or negative infinity, indicating long-term trends (e.g., leveling off or trending upward).

Symmetry

  • Symmetry: Reflects balance in the graph.

  • Even functions: symmetric about the y-axis. Odd functions: symmetric about the origin.

  • Testing for symmetry can be done visually or algebraically.

Chapter 5: Transformations of Functions

Modifying Functions

Transformation Overview

  • Transformations: Techniques to modify parent functions, altering location, orientation, or size of graphs.

Types of Transformations

  • Rigid Transformations: Include translations and reflections; preserve the shape of the graph.

  • Non-Rigid Transformations: Include dilations, altering the size (stretching or compressing) of graphs.

Piecewise Functions

  • Piecewise Functions: Different rules apply to different parts of the domain, useful for modeling complex scenarios (e.g., taxi fare structure).

Combining Functions

  • Function Operations: Add, subtract, multiply, or divide functions to create new functions.

  • Function Composition: Output of one function becomes the input of another.

Chapter 6: Conclusion

Inverse Functions

  • Inverse Functions: Functions that reverse the effect of the original function.

  • Horizontal Line Test: Determines if a function's inverse is also a function; if a horizontal line intersects the graph more than once, the inverse won't be a function.

Reflection on Function Concepts

  • Interconnectedness of function concepts enhances understanding.

  • Encouragement for deeper exploration into cases where multiple outputs exist for a single input, inviting further mathematical inquiry.

Additional Resources

  • Check the show notes for further learning materials related to functions and precalculus.