Graphing and Analyzing Linear Equations: Fresh Take

  • Draw the graph of linear equations by plotting points.
  • Determine the slope based on the steepness and direction of a line.
  • Find the x-intercept and y-intercept of linear equations and graphs.
  • Use formulas to calculate the distances and midpoints between points.

Plotting Points

The Main Idea

  • Origin and Axes:
    • Two perpendicular lines: [latex]x[/latex]-axis (horizontal) and y-axis (vertical)
    • Intersection point of axes: origin, denoted as [latex]0,0[/latex]
  • Quadrants:
    • Plane divided into four quadrants, numbered counterclockwiseThis is an image of an x, y plane with the axes labeled. The upper right section is labeled: Quadrant I. The upper left section is labeled: Quadrant II. The lower left section is labeled: Quadrant III. The lower right section is labeled: Quadrant IV.
  • Ordered Pairs:
    • Points represented as[latex](x,y)[/latex]
    • [latex]x[/latex]-coordinate: horizontal displacement from origin
    • [latex]y[/latex]-coordinate: vertical displacement from origin
  • Plotting Points:
    • Move horizontally by x units, then vertically by y units
    • Positive [latex]x[/latex]: move right; Negative [latex]x[/latex]: move left
    • Positive [latex]y[/latex]: move up; Negative [latex]y[/latex]: move down
  • Special Cases:
    • Points on x-axis: y-coordinate is zero, latex[/latex]
    • Points on y-axis: x-coordinate is zero

 

Plot the following points on the same coordinate plane:

[latex]A(2,3), B(-4,1), C(0,-2), D(-3,-3)[/latex]

You can view the transcript for “Ex: Plotting Points on the Coordinate Plane” here (opens in new window).

Graphing Equations by Plotting Points

The Main Idea

  • Equations in Two Variables:
    • Contain both x and y variables
    • Represented as graphs in a two-dimensional plane
  • Plotting Points:
    • Create a table with x, y, and (x,y) columns
    • Choose strategic x-values for easy calculation
    • Calculate corresponding y-values
    • Plot the resulting ordered pairs (x,y)
  • Connecting Points:
    • If points form a straight line, connect them
    • Not all equations result in straight lines
Construct a table and graph the equation by plotting points: [latex]y=\frac{1}{2}x+2[/latex].

You can view the transcript for “Graphing Equations by Plotting Points – Example 1” here (opens in new window).

Using Intercepts to Plot Lines in the Coordinate Plane

The Main Idea

  • Intercepts Definition:
    • Points where a graph crosses the axes
    • x-intercept: crosses x-axis (y = 0)
    • y-intercept: crosses y-axis (x = 0)
  • Finding x-intercept:
    • Set y = 0 in the equation
    • Solve for x
    • Express as ordered pair (x, 0)
  • Finding y-intercept:
    • Set x = 0 in the equation
    • Solve for y
    • Express as ordered pair (0, y)
  • Graphing Process:
    • Calculate both intercepts
    • Plot the intercept points
    • Draw a line through these points
  • Efficiency:
    • Only two points needed to define a line
    • Quicker than plotting multiple points

 

Find the intercepts of the equation and sketch the graph: [latex]y=-\frac{3}{4}x+3[/latex].

You can view the transcript for “Graphing using x- and y-intercepts | Graphing lines and slope | Algebra Basics | Khan Academy” here (opens in new window).

Slope of a Linear Equation

The Main Idea

  • Definition of Slope:
    • Ratio of vertical change (rise) to horizontal change (run)
    • Measures steepness and direction of a line
    • Formula: [latex]m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}[/latex]
  • Interpretation of Slope:
    • Positive slope: Line rises from left to right
    • Negative slope: Line falls from left to right
    • Zero slope: Horizontal line
    • Undefined slope: Vertical line
  • Slope in Different Contexts:
    • In economics: Rate of change (e.g., marginal cost)
    • In physics: Velocity in distance-time graphs
    • In statistics: Correlation between variables
  • Calculating Slope:
    • Choose any two points on the line
    • Apply the slope formula
    • Simplify the fraction if possible
Find the slope of the line that passes through the points [latex]\left(-2,6\right)[/latex] and [latex]\left(1,4\right)[/latex].

You can view the transcript for “How to find the slope between two points” here (opens in new window).

Distance Formula

The Main Idea

  • Origin and Concept:
    • Derived from the Pythagorean Theorem
    • Calculates the straight-line distance between two points
  • Formula: [latex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/latex] Where [latex](x_1, y_1)[/latex] and [latex](x_2, y_2)[/latex] are the coordinates of two points
  • Geometric Interpretation:
    • Forms a right triangle with the two points and their projections
    • Hypotenuse of this triangle is the distance between points
  • Applications:
    • Navigation and GPS systems
    • Computer graphics and game development
    • Spatial analysis in geography
Find the distance between two points: [latex]\left(1,4\right)[/latex] and [latex]\left(11,9\right)[/latex].

In the following video, we present more worked examples of how to use the distance formula to find the distance between two points in the coordinate plane.

You can view the transcript for “Example: Determine the Distance Between Two Points” here (opens in new window).

Midpoint Formula

The Main Idea

  • Definition:
    • The midpoint is the point that divides a line segment into two equal parts
    • It’s located exactly halfway between the endpoints
  • Formula: For endpoints [latex](x_1, y_1)[/latex] and [latex](x_2, y_2)[/latex], the midpoint [latex]M[/latex] is: [latex]M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)[/latex]
  • Geometric Interpretation:
    • The x-coordinate of the midpoint is the average of the x-coordinates of the endpoints
    • The y-coordinate of the midpoint is the average of the y-coordinates of the endpoints
  • Applications:
    • Finding centers of objects in computer graphics
    • Calculating average positions in physics
    • Determining midpoints of ranges in statistics
Find the midpoint of the line segment with the endpoints [latex]\left(7,-2\right)[/latex] and [latex]\left(9,5\right)[/latex].

Find the midpoint of the line segment with endpoints [latex]\left(-2,-1\right)[/latex] and [latex]\left(-8,6\right)[/latex].

You can view the transcript for “Given two Endpoints Find the Midpoint of a Segment” here (opens in new window).