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7.8 Summary of key concepts

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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 ( [link] )

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

Axis ( [link] )

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 ( [link] )

An   ​ equation   in   one   variable   requires   one   axis . ​ One-dimension . An   equation   in   two   variable   requires   two   axes . Two-dimensions . An   equation   in   three   variable   requires   three   axes . Three-dimensions . ⋅ ⋅ ⋅ An   equation   in   n   variable   requires   n   axes . n -dimensions .

Coordinate system ( [link] )

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

Graphing an equation ( [link] )

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

Uses of a graph ( [link] )

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

Rectangular coordinate system x y -plane ( [link] )

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 x y -plane.

Ordered pairs and points ( [link] )

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 ( [link] )

When graphed, a linear equation produces a straight line.

General form of a linear equation in two variables ( [link] )

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

Graphs, ordered pairs, solutions, and lines ( [link] )

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 ( [link] )

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

Intercept method ( [link] )

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 ( [link] )

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 ( [link] )

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 = y 2 − y 1 x 2 − x 1 = vertical   change horizontal   change

Slope and rise and decline ( [link] )

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

Graphing an equation given in slope-intercept form ( [link] )

An equation written in slope intercept form can be graphed by
  1. Plotting the y -intercept ( 0 , b ) .
  2. Determining another point using the slope, m .
  3. Drawing a line through these two points.

Forms of equations of lines ( [link] )

General   form ¯ Slope-intercept   form ¯ point-slope   from ¯ a x + b y = c y = m x + b y − y 1 = m ( x − x 1 ) To   use   this   form,   the   slope   and   y -intercept are   needed . To   use   this   form,   the slope   and   one   point, or   two   points,   are   needed .

Half-planes and boundary lines ( [link] )

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 ( [link] )

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 ( [link] )

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

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
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Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David Reply
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David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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