Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
Mathematics LibreTexts

11.S: Graphs (Summary)

KEY TERMS

horizontal line The graph of an equation that can be written in the form y = b whose line passes through the y-axis at (0, b).
intercepts of a line Each of the points at which a line crosses the x-axis and the y-axis is called an intercept of the line.
linear equation An equation of the form Ax + By = C, where A and B are not both zero, is called a linear equation in two variables
ordered pair An ordered pair (x, y) gives the coordinates of a point in a rectangular coordinate system. The first number is the x-coordinate. The second number is the y-coordinate.
origin The point (0, 0) is called the origin. It is the point where the the point where the x-axis and y-axis intersect.
quadrants The four areas of a rectangular coordinate system that has been divided by the x-axis and y-axis.
slope of a line The slope of a line is m = \(\frac{rise}{run}\). The rise measures the vertical change and the run measures the horizontal change.
solution to a linear equation in two variables An ordered pair (x, y) is a solution to the linear equation Ax + By = C, if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.
vertical line A vertical line is the graph of an equation that can be written in the form x = a. The line passes through the x-axis at (a, 0).
x-axis The horizontal axis in a rectangular coordinate system.
y-axis The vertical axis on a rectangular coordinate system.

Key Concepts

11.1 Use the Rectangular Coordinate System

  • Sign Patterns of the Quadrants
Quadrant I Quadrant II Quadrant III Quadrant IV
(x,y) (x,y) (x,y) (x,y)
(+,+) (−,+) (−,−) (+,−)
  • Coordinates of Zero
    • Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates (a, 0).
    • Points with a x-coordinate equal to 0 are on the y-axis, and have coordinates (0, b).
    • The point (0, 0) is called the origin. It is the point where the x-axis and y-axis intersect.

11.2 Graphing Linear Equations

  • Graph a linear equation by plotting points.
    1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
    2. Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
    3. Draw the line through the points. Extend the line to fill the grid and put arrows on both ends of the line.
  • Graph of a Linear Equation: The graph of a linear equation ax + by = c is a straight line.
    • Every point on the line is a solution of the equation.
    • Every solution of this equation is a point on this line.

11.3 Graphing with Intercepts

  • Intercepts
    • The x-intercept is the point, (a, 0), where the graph crosses the x-axis. The x-intercept occurs when y is zero.
    • The y-intercept is the point, (0, b), where the graph crosses the y-axis. The y-intercept occurs when y is zero.
    • The x-intercept occurs when y is zero.
    • The y-intercept occurs when x is zero.
  • Find the x and y intercepts from the equation of a line
    • To find the x-intercept of the line, let y = 0 and solve for x.
    • To find the y-intercept of the line, let x = 0 and solve for y.
x y
  0
0  
  • Graph a line using the intercepts
    1. Find the x- and y-intercepts of the line.
      • Let y = 0 and solve for x.
      • Let x = 0 and solve for y.
    2. Find a third solution to the equation.
    3. Plot the three points and then check that they line up.
    4. Draw the line.
  • Choose the most convenient method to graph a line
    1. Determine if the equation has only one variable. Then it is a vertical or horizontal line.
      • x = a is a vertical line passing through the x-axis at a.
      • y = b is a vertical line passing through the y-axis at b.
    2. Determine if y is isolated on one side of the equation. The graph by plotting points. Choose any three values for x and then solve for the corresponding y-values.
    3. Determine if the equation is of the form Ax + By = C, find the intercepts. Find the x- and y-intercepts and then a third point.

11.4 Understand Slope of a Line

  • Find the slope from a graph
    1. Locate two points on the line whose coordinates are integers.
    2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
    3. Count the rise and the run on the legs of the triangle.
    4. Take the ratio of rise to run to find the slope, m = \(\frac{rise}{run}\).
  • Slope of a Horizontal Line
    • The slope of a horizontal line, y = b, is 0.
  • Slope of a Vertical Line
    • The slope of a vertical line, x = a, is undefined.
  • Slope Formula
    • The slope of the line between two points (x1, y1) and (x2, y2) is m = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\).
  • Graph a line given a point and a slope.
    1. Plot the given point.
    2. Use the slope formula to identify the rise and the run.
    3. Starting at the given point, count out the rise and run to mark the second point.
    4. Connect the points with a line.

Contributors