Precalculus Study Guide: Graphs and Functions

Section 1.5: Graphs of Lines

Understanding the Slope and Equation of a Line

This section introduces the fundamental concepts of lines in the coordinate plane, focusing on slope, equations, and their graphical representations.

• Slope of a Line: The slope measures the steepness and direction of a line. It is calculated as the ratio of the change in y to the change in x between two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope Form: The equation of a line given a point $(x_1, y_1)$ and slope $m$ is: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: The equation of a line with slope $m$ and y-intercept $b$ is: $y = mx + b$

• Horizontal and Vertical Lines:

  • Horizontal: $y = c$ (slope $m = 0$)

  • Vertical: $x = c$ (undefined slope)

• Parallel and Perpendicular Lines:

  • Parallel lines have equal slopes.

  • Perpendicular lines have slopes that are negative reciprocals: $m_1 \cdot m_2 = -1$

• Graphing Linear Equations: Plot points and use slope to draw the line.

Example:

Find the equation of a line passing through $(2, 3)$ with slope $4$:

$y - 3 = 4(x - 2)$

Section 2.1: Functions and Their Graphs

Definition and Representation of Functions

This section explores the concept of functions, their notation, and how to represent them graphically and algebraically.

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

• Function Notation: $f(x)$ denotes the output for input $x$.

• Domain and Range:

  • Domain: Set of all possible input values.

  • Range: Set of all possible output values.

• Graph of a Function: The set of points $(x, f(x))$ in the coordinate plane.

• Vertical Line Test: A graph represents a function if no vertical line intersects it more than once.

Example:

Is $y = x^2$ a function? Yes, because each $x$ has one $y$ value.

Section 2.2: Graphs of Functions

Interpreting and Sketching Function Graphs

This section focuses on identifying and analyzing the graphs of functions, including transformations and key features.

• Identifying Graphs: Recognize common function shapes (linear, quadratic, etc.).

• Transformations:

  • Vertical shifts: $f(x) + c$

  • Horizontal shifts: $f(x - c)$

  • Reflections: $-f(x)$ or $f(-x)$

  • Stretching/Compressing: $a f(x)$

• Key Features:

  • Intercepts (where graph crosses axes)

  • Maximum and minimum points

  • Intervals of increase/decrease

Example:

Sketch $f(x) = x^2$ and $f(x) = x^2 + 3$ (vertical shift up by 3).

Section 2.3: Linear and Quadratic Functions

Properties and Applications of Linear and Quadratic Functions

This section covers the characteristics, equations, and applications of linear and quadratic functions.

• Linear Functions:

  • General form: $f(x) = mx + b$

  • Graph is a straight line.

• Quadratic Functions:

  • General form: $f(x) = ax^2 + bx + c$

  • Graph is a parabola.

  • Vertex: $x = -\frac{b}{2a}$

  • Axis of symmetry: $x = -\frac{b}{2a}$

• Maximum/Minimum:

  • If $a > 0$, parabola opens upward (minimum at vertex).

  • If $a < 0$, parabola opens downward (maximum at vertex).

• Applications: Used in modeling real-world situations such as projectile motion.

Example:

Find the vertex of $f(x) = 2x^2 - 4x + 1$:

$x = -\frac{-4}{2 \cdot 2} = 1$

Section 2.4: Library of Functions and Graphs

Common Functions and Their Graphs

This section introduces basic functions frequently used in precalculus and their graphical representations.

• Linear Function: $f(x) = x$

• Quadratic Function: $f(x) = x^2$

• Cubic Function: $f(x) = x^3$

• Square Root Function: $f(x) = \sqrt{x}$

• Absolute Value Function: $f(x) = |x|$

• Graphing: Recognize the shape and key points of each function.

Example:

Graph $f(x) = |x|$; it forms a 'V' shape with vertex at $(0, 0)$.

Practice Problems and Objectives

Review and Practice for Mastery

Practice problems are provided for each section to reinforce understanding and prepare for exams. These include concept checks, vocabulary, and application exercises.

• Section 1.5: Slope, equations of lines, graphing, parallel/perpendicular lines

• Section 2.1: Functions, domain/range, vertical line test

• Section 2.2: Graphs, transformations, key features

• Section 2.3: Linear/quadratic functions, maximum/minimum, applications

• Section 2.4: Library of functions, graphing basic functions

Additional info: Practice problems are referenced by textbook section and include odd/even numbered exercises, concept and vocabulary checks, and application questions. These are essential for exam preparation and mastery of precalculus fundamentals.