# Cheat sheet

## 1 Copyright

This work is distributed under the CC BY-NC-SA license. It is derived, with changes, from a work of William Pierpoint. The changes are copyright Jonathan Fine.

## 2 About

This is one quarter of a College Algebra Quick Reference (cheat sheet) prepared by William Pierpoint (California State University, Channel Islands).

The original document (PDF) was prepared using Microsoft Word in 2013, and published as a PDF. This derived work has be prepared using https://quarto.org/ with the mathematics rendered by https://www.mathjax.org/

For more information about the original visit https://www.stitz-zeager.com/.

## 3 Creative Commons

The original documents seems to be licensed under Creative Commons BY-NC-SA. That license allows sharing and adaption provided

- The original work is properly attributed.
- It is indicated if changes are made from the original work.
- It is not suggested that the licensor endorses the derived work.
- The new work is not used for any commercial purpose.
- The new work must be distributes under the same license as the original.

## 4 Set notation

Interval Notation | Set-Builder Notation |
---|---|

\((a, b)\) | \(\{ x \mid a < x < b \}\) |

\([a, b]\) | \(\{ x \mid a \leq x \leq b \}\) |

\([a, b)\) | \(\{ x \mid a \leq x < b \}\) |

\((a, b]\) | \(\{ x \mid a < x \leq b \}\) |

\((a, \infty)\) | \(\{ x \mid a < x \}\) |

\([a, \infty)\) | \(\{ x \mid a \leq x \}\) |

\((-\infty, b)\) | \(\{ x \mid x < b \}\) |

\((-\infty, b]\) | \(\{ x \mid x \leq b \}\) |

## 5 Set operations

Operation | Elements | Logic |
---|---|---|

Union \(\cup\) | All | OR |

Intersection \(\cap\) | Common | AND |

## 6 Coordinate Plane Quadrants

II | I |

III | IV |

## 7 Distance and Midpoint Formulas

If \(P_1 = (x_1, y_1)\) and \(P_2 = (x_2, y_2)\) are two points, the distance between them is \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2)} \] and the midpoint coordinates are \[ M = \left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}\right) \]

- Latex:
`d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2)}`

- Latex:
`M = \left(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2}\right)`

## 8 Intercepts of an Equation

\(x\)-intercepts | Set \(y=0\); solve for \(x\) |

\(y\)-intercepts | Set \(x=0\); solve for \(y\) |

## 9 Symmetry of the Graph of an Equation

Type | Mathematical | Geometrical |
---|---|---|

\(x\)-axis | Unchanged when \(y\) replaced by \(-y\) | Unchanged when reflected about \(x\)-axis |

\(y\)-axis | Unchanged when \(x\) replaced by \(-x\) | Unchanged when reflected about \(y\)-axis |

origin | Unchanged when \(x\) replaced by \(-y\) & \(x\) replaced by \(-x\) | Unchanged when rotated \(180\degree\) about origin |

## 10 Function Notation \(y=f(x)\)

Domain | Set of all valid \(x\) |

Range | Set of all valid \(y\) |

## 11 Function Arithmetic

\((f+g)(x) = f(x) + g(x)\)

\((f-g)(x) = f(x) - g(x)\)

\((fg)(x) = f(x)g(x)\)

\(\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}\)

Latex:

`\left(\dfrac{f}{g}\right)(x) = \dfrac{f(x)}{g(x)}`

## 12 Transformations of Graphs of Functions

[This is long and complicated. It’s being skipped for now.]

## 13 Properties of Equality

- [In PDF:] If \(a=b\) then \(a+c=a+c\) and \(a-c = a-c\)
- [Should be:] If \(a=b\) then \(a+c=b+c\) and \(a-c = b-c\)
- If \(a=b\) and \(c\neq0\) then \(ac=bc\) and \(\dfrac{a}{c}=\dfrac{b}{c}\)

## 14 Properties of Inequalities

If \(a<b\) then \(a+c < b+c\) and \(a-c < b-c\)

If \(a<b\) and \(c>0\) then \(ac < bc\) and \(\dfrac{a}{c} < \dfrac{b}{c}\)

If \(a<b\) and \(c<0\) then \(ac > bc\) and \(\dfrac{a}{c} > \dfrac{b}{c}\)

Latex:

`\dfrac{a}{c} < \dfrac{b}{c}`