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Summary of Key Concepts

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. In this chapter the student is shown how graphs provide information that is not always evident from the equation alone. The chapter begins by establishing the relationship between the variables in an equation, the number of coordinate axes necessary to construct its graph, and the spatial dimension of both the coordinate system and the graph. Interpretation of graphs is also emphasized throughout the chapter, beginning with the plotting of points. The slope formula is fully developed, progressing from verbal phrases to mathematical expressions. The expressions are then formed into an equation by explicitly stating that a ratio is a comparison of two quantities of the same type (e.g., distance, weight, or money). This approach benefits students who take future courses that use graphs to display information. The student is shown how to graph lines using the intercept method, the table method, and the slope-intercept method, as well as how to distinguish, by inspection, oblique and horizontal/vertical lines. This module presents a summary of the key concepts of the chapter "Graphing Linear Equations and Inequalities in One and Two Variables".

Summary of Key Concepts

Graph of a Function ((Reference))

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

Axis ((Reference))

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 ((Reference))

Anequationinonevariablerequiresoneaxis. One-dimension. Anequationintwovariablerequirestwoaxes. Two-dimensions. Anequationinthreevariablerequiresthreeaxes. Three-dimensions. Anequationinnvariablerequiresnaxes. n-dimensions. Anequationinonevariablerequiresoneaxis. One-dimension. Anequationintwovariablerequirestwoaxes. Two-dimensions. Anequationinthreevariablerequiresthreeaxes. Three-dimensions. Anequationinnvariablerequiresnaxes. n-dimensions.

Coordinate System ((Reference))

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

Graphing an Equation ((Reference))

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

Uses of a Graph ((Reference))

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

Rectangular Coordinate System xy xy -Plane ((Reference))

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

Ordered Pairs and Points ((Reference))

For each ordered pair ( a,b ), ( 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 ) ( a,b ) of real numbers.

Graphs of Linear Equations ((Reference))

When graphed, a linear equation produces a straight line.

General Form of a Linear Equation in Two Variables ((Reference))

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

Graphs, Ordered Pairs, Solutions, and Lines ((Reference))

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 ((Reference))

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

Intercept Method ((Reference))

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 x-axis and the y-axis y-axis .

Slanted, Vertical, and Horizontal Lines ((Reference))

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 x=a graphs as a vertical line passing through a a on the x-axis x-axis .
y=b y=b graphs as a horizontal line passing through b b on the y-axis y-axis .

Slope of a Line ((Reference))

The slope of a line is a measure of the line’s steepness. If ( x 1 , y 1 ) ( x 1 , y 1 ) and ( x 2 , y 2 ) ( 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= y 2 y 1 x 2 x 1 = verticalchange horizontalchange m= y 2 y 1 x 2 x 1 = verticalchange horizontalchange

Slope and Rise and Decline ((Reference))

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

Graphing an Equation Given in Slope-Intercept Form ((Reference))

An equation written in slope intercept form can be graphed by

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

Forms of Equations of Lines ((Reference))

Generalform ¯ Slope-interceptform ¯ point-slopefrom ¯ ax+by=c y=mx+b y y 1 =m(x x 1 ) Tousethisform,the slopeandy-intercept areneeded. Tousethisform,the slopeandonepoint, ortwopoints,areneeded. Generalform ¯ Slope-interceptform ¯ point-slopefrom ¯ ax+by=c y=mx+b y y 1 =m(x x 1 ) Tousethisform,the slopeandy-intercept areneeded. Tousethisform,the slopeandonepoint, ortwopoints,areneeded.

Half-Planes and Boundary Lines ((Reference))

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 ((Reference))

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 ((Reference))

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

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