7.9: Summary of Key Concepts

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Summary of Key Concepts

Graph of a Function

The geometric representation (picture) of the solutions to an equation is called the graph of the equation.

Axis

An axis is the most basic structure of a graph. In mathematics, the number line is used as an axis.

Number of Variables and the Number of Axes

An equation in one variable requires one axis	One-dimension
An equation in two variables requires two axes.	Two-dimensions
An equation in three variables requires three axes.	Three-dimensions.
An equation in \(n\) variables requires \(n\) axes	\(n\)-dimensions.

Coordinate System

A system of axes that is constructed for graphing an equation is called a coordinate system.

Graphing an Equation

The phrase graphing an equation is interpreted as meaning geometrically locating the solutions to that equation.

Uses of a Graph

A graph may reveal information that may not be evident from the equation.

Rectangular Coordinate System \(xy\)-Plane

A rectangular coordinate system is constructed by placing two number lines at 90∘ angles. These lines form a plane that is referred to as the \(xy\)-plane.

Ordered Pairs and Points

For each ordered pair \((a,b)\), there exists a unique point in the plane, and for each point in the plane we can associate a unique ordered pair \((a,b)\) of real numbers.

Graphs of Linear Equations

When graphed, a linear equation produces a straight line.

General Form of a Linear Equation in Two Variables

The general form of a linear equation in two variables is \(ax+by=c\), where \(a\) and \(b\) are not both \(0\).

Graphs, Ordered Pairs, Solutions, and Lines

The graphing of all ordered pairs that solve a linear equation in two variables produces a straight line.
The graph of a linear equation in two variables is a straight line.
If an ordered pair is a solution to a linear equation in two variables, then it lies on the graph of the equation.
Any point (ordered pair) that lies on the graph of a linear equation in two variables is a solution to that equation.

Intercept

An intercept is a point where a line intercepts a coordinate axis.

Intercept Method

The intercept method is a method of graphing a linear equation in two variables by finding the intercepts, that is, by finding the points where the line crosses the \(x\)-axis and the \(y\)-axis.

Slanted, Vertical, and Horizontal Lines

An equation in which both variables appear will graph as a slanted line.
A linear equation in which only one variable appears will graph as either a vertical or horizontal line.
\(x=a\) graphs as a vertical line passing through \(a\) on the \(x\)-axis.
\(y=b\) graphs as a horizontal line passing through \(b\) on the \(y\)-axis.

Slope of a Line

The slope of a line is a measure of the line’s steepness. If \((x_1,y_1)\) and \((x_2,y_2)\) are any two points on a line, the slope of the line passing through these points can be found using the slope formula.

\(m = \dfrac{y_2-y_1}{x_2-x_1} = \dfrac{\text{ vertical change }}{\text{ horizontal change }}\)

Slope and Rise and Decline

Moving left to right, lines with positive slope rise, and lines with negative slope decline.

Graphing an Equation Given in Slope-Intercept Form

An equation written in slope-intercept form can be graphed by

Plotting the \(y\)-intercept \((0,b)\).
Determining another point using the slope, \(m\).
Drawing a line through these two points.

Forms of Equations of Lines

General Form: \(ax + by + c\)

Slope-Intercept Form: \(y = mx + b\)
To use this form, the slope and \(y\)-intercept are needed

Point-Slope Form: \(y - y_1 = m(x - x_1)\)
To use this form, the slope and one points, or two points, are needed.

Half-Planes and Boundary Lines

A straight line drawn through the plane divides the plane into two half-planes. The straight line is called a boundary line.

Solution to an Inequality in Two Variables

A solution to an inequality in two variables is a pair of values that produce a true statement when substituted into the inequality.

Location of Solutions to Inequalities in Two Variables

All solutions to a linear inequality in two variables are located in one, and only one, half-plane.