# 11.S: Graphs (Summary)

- Page ID
- 5045

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)

## 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 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 = \(\dfrac{rise}{run}\). 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**.- Find three points whose coordinates are solutions to the equation. Organize them in a table.
- Plot the points on a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
- 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 x 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**- Find the x- and y-intercepts of the line.
- Let y = 0 and solve for x.
- Let x = 0 and solve for y.

- Find a third solution to the equation.
- Plot the three points and then check that they line up.
- Draw the line.

- Find the x- and y-intercepts of the line.
**Choose the most convenient method to graph a line**- 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 horizontal line passing through the y-axis at b.

- 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.
- 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.

- Determine if the equation has only one variable. Then it is a vertical or horizontal line.

### 11.4 Understand Slope of a Line

**Find the slope from a graph**- Locate two points on the line whose coordinates are integers.
- Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
- Count the rise and the run on the legs of the triangle.
- Take the ratio of rise to run to find the slope, m = \(\dfrac{rise}{run}\).

**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 (x
_{1}, y_{1}) and (x_{2}, y_{2}) is m = \(\dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}\).

- The slope of the line between two points (x
**Graph a line given a point and a slope.**- Plot the given point.
- Use the slope formula to identify the rise and the run.
- Starting at the given point, count out the rise and run to mark the second point.
- Connect the points with a line.

## Contributors

Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."