7.9: Summary of Key Concepts
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.