Slope and Graphing in Coordinate System

Rectangular Coordinate System

Slope Of A Line

• Definition: The slope (m) measures the steepness or incline of a line.
• Formula: m = (y₂ - y₁) / (x₂ - x₁)
• Types of Slope:
  • Positive slope: line rises as it moves right.
  • Negative slope: line falls as it moves right.
  • Zero slope: horizontal line.
  • Undefined slope: vertical line.

Graphing Linear Equations

• Standard Form: Ax + By = C, where A, B, and C are constants.
• Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
• Steps to Graph:
  1. Identify the y-intercept (b).
  2. Use the slope (m) to find another point.
  3. Plot both points and draw a straight line through them.

Cartesian Coordinates

• Definition: A system for describing points in a plane using pairs of numbers.
• Components:
  • X-axis: horizontal axis.
  • Y-axis: vertical axis.
• Point Notation: Each point is expressed as (x, y), where:
  • x = horizontal distance from the origin.
  • y = vertical distance from the origin.

Quadrants

• Definition: The Cartesian plane is divided into four quadrants.
• Quadrant Descriptions:
  1. Quadrant I: (x > 0, y > 0) - both coordinates positive.
  2. Quadrant II: (x < 0, y > 0) - x negative, y positive.
  3. Quadrant III: (x < 0, y < 0) - both coordinates negative.
  4. Quadrant IV: (x > 0, y < 0) - x positive, y negative.

Solving For Equation of a Line

• Using Two Points:

  1. Determine the slope (m) using two points (x₁, y₁) and (x₂, y₂).
  2. Use point-slope form: y - y₁ = m(x - x₁).
  3. Rearrange to slope-intercept form (y = mx + b) or standard form (Ax + By = C).

• Using One Point and Slope:

  1. If given a slope (m) and a point (x₀, y₀), use: y - y₀ = m(x - x₀).
  2. Rearrange as necessary for different forms.

• Y-Intercept: If the equation is in slope-intercept form, b directly gives the y-intercept.

Slope of a Line

• The slope (m) represents a line's steepness.
• Calculated by: m = (y₂ - y₁) / (x₂ - x₁)
• A positive slope indicates a line rising from left to right.
• A negative slope indicates a line falling from left to right.
• A horizontal line has a slope of zero.
• A vertical line has an undefined slope.

Graphing Linear Equations

• The standard form of a linear equation is Ax + By = C.
• The slope-intercept form is y = mx + b, where b represents the y-intercept.
• To graph a linear equation:
  • Locate the y-intercept (b) on the y-axis.
  • Utilize the slope (m) to find another point on the line.
  • Plot both points and draw a straight line through them.

Cartesian Coordinates

• Provides a system for identifying points in a plane using number pairs.
• The x-axis is horizontal, and the y-axis is vertical.
• Each point is denoted as (x, y):
  • x represents the horizontal distance from the origin.
  • y represents the verticaldistance from the origin.

Quadrants

• The Cartesian plane is divided into four quadrants.
• Quadrant I: Both x and y coordinates are positive (x > 0, y > 0).
• Quadrant II: x is negative and y is positive (x < 0, y > 0).
• Quadrant III: Both x and y coordinates are negative (x < 0, y < 0).
• Quadrant IV: x is positive and y is negative (x > 0, y < 0).

Solving for the Equation of a Line

• Using two points (x₁, y₁) and (x₂, y₂):
  • Calculate the slope (m) using the formula.
  • Apply the point-slope form: y - y₁ = m(x - x₁).
  • Rearrange the equation into either slope-intercept form (y = mx + b) or standard form (Ax + By = C).
• Using one point (x₀, y₀) and slope (m):
  • Employ the point-slope form: y - y₀ = m(x - x₀).
  • Rearrange the equation as needed for different forms.
• The y-intercept (b) can be directly identified if the equation is in slope-intercept form.