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.

Explore the concepts of slope and graphing linear equations in the rectangular coordinate system. This quiz covers definitions, formulas, and types of slope, as well as methods for graphing linear equations in standard and slope-intercept forms.