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3.1: Graphing Equations by Hand
https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(Arnold)/03%3A_Introduction_to_Graphing/3.01%3A_Graphing_Equations_by_Hand
However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of th...However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of the line shown in Figure $\PageIndex{11}$ . However, the collection of points in Figure $\PageIndex{21}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x^2 −7$ , the result would be the curve (called a parabola) shown in Figure $\PageIndex{22}$ .
4.4: Cartesian Products
https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/A_Spiral_Workbook_for_Discrete_Mathematics_(Kwong)/04%3A_Sets/4.04%3A_Cartesian_Products
For any sets $A$ , $B$ , and $C$ , we have \[\begin{array}{r c l} A \times (B \cup C) &=& (A \times B) \cup (A \times C), \\ A \times (B \cap C) &=& (A \times B) \cap (A \times C), \\ A \times (B -...For any sets $A$ , $B$ , and $C$ , we have $\begin{array}{r c l} A \times (B \cup C) &=& (A \times B) \cup (A \times C), \\ A \times (B \cap C) &=& (A \times B) \cap (A \times C), \\ A \times (B - C) &=& (A \times B) - (A \times C).\end{array} \nonumber$
1.4: Graphing
https://math.libretexts.org/Courses/Mt._San_Jacinto_College/Ideas_of_Mathematics/01%3A_Number_Sense/1.04%3A_Graphing
René Descartes (1596-1650) was a French philosopher and mathematician. As a philosopher, he is famous for the saying “Cogito ergo sum” (“I think, therefore I am”), and his writings led many to conside...René Descartes (1596-1650) was a French philosopher and mathematician. As a philosopher, he is famous for the saying “Cogito ergo sum” (“I think, therefore I am”), and his writings led many to consider him the Father of Modern Philosophy. Even today, a number of his writings are standard faire in university philosophy departments. However, it is Descartes’ work in mathematics that form the basis for this chapter, particularly his invention of the Cartesian Coordinate System which bears his name
6.5: Inverse Functions
https://math.libretexts.org/Bookshelves/Mathematical_Logic_and_Proof/Book%3A_Mathematical_Reasoning__Writing_and_Proof_(Sundstrom)/06%3A_Functions/6.05%3A_Inverse_Functions
This also means that if we start with a subset $f$ of $A \times B$ that satisfies conditions in Equation $\ref{6.5.1}$ and $\ref{6.5.2}$ , then we can consider $f$ to be a function from $A$ to \(B\...This also means that if we start with a subset $f$ of $A \times B$ that satisfies conditions in Equation $\ref{6.5.1}$ and $\ref{6.5.2}$ , then we can consider $f$ to be a function from $A$ to $B$ by using $b = f(a)$ whenever $(a, b)$ is in $f$ . In the situation where $f: A \to B$ is a bijection and $f^{-1}$ is a function from $B$ to $A$ , we can write $f^{-1}: B \to A$ .
6.1: Paired Data and Graphing Ordered Pairs
https://math.libretexts.org/Courses/Western_Technical_College/PrePALS_PreAlgebra/06%3A_Equations_in_Two_Variables_and_Graphing/6.01%3A_Paired_Data_and_Graphing_Ordered_Pairs
Let’s begin with the concept of an ordered pair of whole numbers.
2.1.4: Graphing Equations by Hand
https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/MAT_149%3A_Topics_in_Finite_Mathematics_(Holz)/02%3A_Functions/2.01%3A_Functions/2.1.04%3A_Graphing_Equations_by_Hand
However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of th...However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of the line shown in Figure $\PageIndex{11}$ . However, the collection of points in Figure $\PageIndex{21}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x^2 −7$ , the result would be the curve (called a parabola) shown in Figure $\PageIndex{22}$ .
1.1: Graphing Equations by Hand
https://math.libretexts.org/Courses/Highline_College/Math_084__Intermediate_Algebra_Foundations_for_Soc_Sci_Lib_Arts_and_GenEd/01%3A_Introduction_to_Graphing_-_By_Hand_and_Calculator/1.01%3A_Graphing_Equations_by_Hand
However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of th...However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of the line shown in Figure $\PageIndex{11}$ . However, the collection of points in Figure $\PageIndex{21}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x^2 −7$ , the result would be the curve (called a parabola) shown in Figure $\PageIndex{22}$ .
7.1: Graphing by Hand in the Two-Dimensional Rectangular Coordinate System
https://math.libretexts.org/Courses/Honolulu_Community_College/Math_75X%3A_Introduction_to_Mathematical_Reasoning_(Kearns)/07%3A_Introduction_to_Graphing/7.00%3A_Graphing_by_Hand_in_the_Two-Dimensional_Rectangular_Coordinate_System
However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of th...However, the collection of points plotted in Figure $\PageIndex{10}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x + 1$ , we would get the graph of the line shown in Figure $\PageIndex{11}$ . However, the collection of points in Figure $\PageIndex{21}$ suggest that if we were to plot the remainder of the points that satisfy the equation $y = x^2 −7$ , the result would be the curve (called a parabola) shown in Figure $\PageIndex{22}$ .
1.2: The Cartesian Product
https://math.libretexts.org/Courses/Borough_of_Manhattan_Community_College/MAT_310_Bridge_to_Advanced_Mathematics/01%3A_Sets/1.02%3A_The_Cartesian_Product
The Cartesian product of two sets A and B is another set, denoted as $A \times B$ and defined as $A \times B = \{(a, b) : a \in A, b in B\}$ . It is even possible for one factor of a Cartesian prod...The Cartesian product of two sets A and B is another set, denoted as $A \times B$ and defined as $A \times B = \{(a, b) : a \in A, b in B\}$ . It is even possible for one factor of a Cartesian product to be a Cartesian product itself, as in $\mathbb{R} \times (\mathbb{N} \times \mathbb{Z}) = \{(x, (y, z)) : x \in \mathbb{R}, (y, z) \in \mathbb{N} \times \mathbb{Z}\}$ . $A_{n} = A \times A \times \dots \times A = \{(x_{1}, x_{2}, \dots , x_{n}) : x_{1}, x_{2}, \dots ,x_{n} \in A\}$ .
8.2: The Cartesian Coordinate System
https://math.libretexts.org/Bookshelves/PreAlgebra/Prealgebra_(Arnold)/08%3A_Graphing/8.02%3A_The_Cartesian_Coordinate_System
Let’s begin with the concept of an ordered pair of whole numbers.
2.2: Ordered Pairs
https://math.libretexts.org/Bookshelves/Applied_Mathematics/Calculus_for_Business_and_Social_Sciences_Corequisite_Workbook_(Dominguez_Martinez_and_Saykali)/02%3A_Cartesian_Coordinate_System/2.02%3A_Ordered_Pairs
Ordered pairs are pairs of numbers used to locate a point in the rectangular coordinate plane and written in the form (x,y) , where x is the x-coordinate and y is the y-coordinate.

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