Notes on Graphing Equations

Overview

• Focus on graphing various types of equations:
  • Linear equations (including inequalities)
  • Quadratic equations
  • Transformations
  • Radical functions
  • Cubic functions
  • Absolute value equations
  • Rational expressions
  • Exponential equations
  • Logarithmic equations

Graphing Linear Equations

Basic Example: y = 2x + 3

1. Create a Table of Values:
   • Choose x values: -1, 0, 1
   • Calculate corresponding y values:
     • For x = -1: y = 2(-1) + 3 = 1
     • For x = 0: y = 3
     • For x = 1: y = 5
   • Plot points (-1, 1), (0, 3), and (1, 5).
   • Optional point: x = -2, y = -1.
2. Connect Points with a line.

Slope-Intercept Form: y = mx + b

• y-intercept (b): Starting point.
• Slope (m): Rise over run.
  • Example: For y = -2x + 1, start at (0, 1) and move down 2 units for every 1 unit to the right.

Standard Form: Ax + By = C

1. Find x and y intercepts:
   • Example: 2x + 3y = 6
   • x-intercept (set y=0): x = 3
   • y-intercept (set x=0): y = 2
2. Plot intercepts and connect.

Graphing Inequalities

• Convert to slope-intercept form.
• Use dashed lines for inequalities (e.g., y > ...).
• Shade the appropriate region based on the inequality.

Quadratic Equations

Example: y = x^2 + 2x + 3

1. Find Vertex: x = -b/(2a).
2. Create a Table: Pick points around the vertex to find y-values.
3. Plot and Connect.

Vertex Form: y = a(x-h)^2 + k

• Vertex (h, k) can be easily identified from the equation.

Reflection and Transformation

• A negative sign indicates reflection across the x-axis.
• Changing coefficients alters the steepness and direction.

Absolute Value Functions

• General shape is a V.
• Vertex identifies the point of symmetry.

Radical Functions

Example: y = √x

• Plots a curve starting from the origin.
• Shifts to left/right and up/down based on the equation format.

Exponential Functions

Example: y = 2^(x + 3) - 1

1. Find key points by setting the exponent equal to certain values (e.g., 0, 1).
2. Determine horizontal asymptotes.

Logarithmic Functions

Example: y = log₂(x + 1) - 3

1. Set the inside equal to 0, 1, and the base for vertical asymptotes.
2. Plot key points and asymptotes.

Summary of Key Concepts

• Intercepts: Useful for plotting linear and standard form equations.
• Vertex: Important in quadratics and absolute value functions.
• Asymptotes: Critical for rational, exponential, and logarithmic functions.
• Inequalities: Require dashed or solid lines and shading.

Final Note

These principles apply broadly to various equations encountered in algebra, trigonometry, pre-calculus, and calculus.