Linear Equations, Coordinate Geometry, and Graphing Foundations

Module Overview and Completion Status

• Grade Earned: $48\%$
• Completion Rate: $52\%$
• Late Work Policy: No late credit permitted.
• Core Topics Included:
  • Plotting Points
  • Identifying Points (Level 1)
  • Identifying Points (Level 2)
  • Finding the Slope Graphically
  • Finding the Slope from Points
  • Graph Line from Slope/Y-intercept

Coordinate Plane Fundamentals and Plotting Points

• The Cartesian coordinate system consists of two perpendicular real number lines that intersect at a central reference point called the origin $(0, 0)$.
• Horizontal Axis ($x$-axis): Measures horizontal displacement from the origin. Values to the right of the origin are positive ($x > 0$), and values to the left are negative ($x < 0$).
• Vertical Axis ($y$-axis): Measures vertical displacement from the origin. Values above the origin are positive ($y > 0$), and values below are negative ($y < 0$).
• Division of Quadrants: The two axes divide the coordinate plane into four distinct regions called quadrants, numbered counterclockwise using Roman numerals:
  • Quadrant I: Both coordinates are positive, $(+,+)$.
  • Quadrant II: The $x$-coordinate is negative and the $y$-coordinate is positive, $(-,+)$.
  • Quadrant III: Both coordinates are negative, $(-,-)$.
  • Quadrant IV: The $x$-coordinate is positive and the $y$-coordinate is negative, $(+,-)$.
• Ordered Pairs: Every point on the plane is uniquely identified by an ordered pair $(x, y)$:
  • Abscissa ($x$): The first value, representing horizontal position.
  • Ordinate ($y$): The second value, representing vertical position.
• Step-by-Step Procedure for Plotting Points:
  1. Begin at the origin $(0, 0)$.
  2. Locate the $x$-coordinate: Move horizontally along the $x$-axis $|x|$ units right if $x > 0$, or $|x|$ units left if $x < 0$.
  3. Locate the $y$-coordinate: From the current position on the $x$-axis, move vertically parallel to the $y$-axis $|y|$ units up if $y > 0$, or $|y|$ units down if $y < 0$.
  4. Draw a point at the final position and label it with its coordinates $(x, y)$.

Identifying Points (Level 1 and Level 2)

• Level 1 Identification (Standard Integer Grids):

  • Involves reading point locations directly from standard coordinate grids where each grid line represents $1$ unit.
  • To identify the coordinates of a plotted point:
  1. Project vertically from the point to the $x$-axis to determine the $x$-coordinate.
  2. Project horizontally from the point to the $y$-axis to determine the $y$-coordinate.
  3. Write the point in standard ordered pair notation $(x, y)$.
  • Points on Axes:
  • Any point located directly on the $x$-axis has a $y$-coordinate equal to zero, taking the general form $(x, 0)$.
  • Any point located directly on the $y$-axis has an $x$-coordinate equal to zero, taking the general form $(0, y)$.

• Level 2 Identification (Non-Standard Scales and Rational Coordinates):

  • Involves identifying points on grids with non-unit axis increments or points located between integer grid lines.
  • Scale Factor Evaluation: Always calculate the scale per grid division before reading coordinates:     $\text{Scale per grid line} = \frac{\text{Value of major tick mark}}{\text{Number of grid intervals}}$
  • Non-Integer Coordinates: Points positioned midway between grid lines represent rational numbers or decimals (e.g., $0.5$, $1.5$, or fractions such as $\frac{1}{2}$, $\frac{3}{4}$).
  • Quadrant-Based Sign Verification: Verify that the signs of both coordinates match the expected quadrant rules:
  • Quadrant I: $(x > 0, y > 0)$
  • Quadrant II: $(x < 0, y > 0)$
  • Quadrant III: $(x < 0, y < 0)$
  • Quadrant IV: $(x > 0, y < 0)$

Finding the Slope Graphically

• Definition of Slope: The slope $m$ of a non-vertical line measures its steepness and direction, defined as the ratio of vertical change (rise) to horizontal change (run).

• Graphical Slope Formula:   $m = \frac{\text{Rise}}{\text{Run}}$

• Step-by-Step Procedure for Determining Slope from a Graph:

  1. Identify two distinct points on the line that sit exactly at grid intersections, $(x_1, y_1)$ and $(x_2, y_2)$.
  2. Select the leftmost point as the starting point to ensure a positive run.
  3. Calculate the Rise: Count the vertical grid units required to move from the starting point's $y$-level to the ending point's $y$-level.
  • Upward movement yields a positive rise ($+\text{Rise}$).
  • Downward movement yields a negative rise ($-\text{Rise}$).
  1. Calculate the Run: Count the horizontal grid units moving left-to-right from the starting point's $x$-level to the ending point's $x$-level ($+\text{Run}$).
  2. Form the fraction $\frac{\text{Rise}}{\text{Run}}$ and reduce it to simplest form.

• Four Classifications of Slope:

  • Positive Slope ($m > 0$): The line rises from left to right.
  • Negative Slope ($m < 0$): The line falls from left to right.
  • Zero Slope ($m = 0$): The line is horizontal ($y = c$), where vertical change is zero ($\text{Rise} = 0$).
  • Undefined Slope ($m = \text{undefined}$): The line is vertical ($x = c$), where horizontal change is zero ($\text{Run} = 0$), causing division by zero.

Finding the Slope from Points

• Algebraic Definition: For any two distinct points $(x_1, y_1)$ and $(x_2, y_2)$ on a line where $x_1 \neq x_2$, the slope $m$ is given by the ratio of the change in $y$ to the change in $x$:   $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Alternative Notation:   $m = \frac{\Delta y}{\Delta x}$   where $\Delta y = y_2 - y_1$ represents the vertical change, and $\Delta x = x_2 - x_1$ represents the horizontal change.

• Step-by-Step Calculation Procedure:

  1. Label the given points explicitly as $(x_1, y_1)$ and $(x_2, y_2)$.
  2. Substitute the coordinates directly into the slope formula $m = \frac{y_2 - y_1}{x_2 - x_1}$.
  3. Simplify the numerator ($y_2 - y_1$) and denominator ($x_2 - x_1$), carefully tracking negative signs.
  4. Reduce the resulting fraction to its lowest terms.

• Key Algebraic Special Cases:

  • Equal $y$-coordinates ($y_1 = y_2$):     $m = \frac{0}{x_2 - x_1} = 0$     This indicates a horizontal line.
  • Equal $x$-coordinates ($x_1 = x_2$):     $m = \frac{y_2 - y_1}{0} = \text{undefined}$     This indicates a vertical line.

Graph Line from Slope and Y-Intercept

• Slope-Intercept Form Equation:   $y = m x + b$   where $m$ is the slope of the line and $b$ is the $y$-intercept value, corresponding to the point $(0, b)$.

• Essential Components:

  • Slope ($m$): Dictates the rate of change and direction of the line. Express as a ratio $\frac{\text{rise}}{\text{run}}$. If $m$ is an integer, rewrite it as $\frac{m}{1}$.
  • Y-Intercept ($b$): The point $(0, b)$ where the line intersects the vertical $y$-axis.

• Step-by-Step Graphing Procedure:

  1. Plot the $y$-intercept $(0, b)$ on the vertical axis.
  2. Use the slope $m = \frac{\text{rise}}{\text{run}}$ to locate a second point:
  • Move vertically from $(0, b)$ by the amount specified in the numerator (up if positive, down if negative).
  • Move horizontally to the right by the amount specified in the denominator.
  • Plot the new point at $(0 + \text{run}, b + \text{rise})$.
  1. Repeat the slope count from the new point to plot a third point to verify alignment.
  2. Draw a straight line through all plotted points using a straightedge, extending across the entire grid with arrows at both ends to indicate infinite continuation.