"A coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space."
Graphing linear equations involves plotting points on a coordinate plane to visualize linear equations.
Variables and Equations: Understanding the use of variables and solving equations is fundamental to graphing linear equations. You need to know how to solve equations and understand the concept of a variable.
Coordinate Grid: A grid system used for plotting points and graphing equations. You need to be able to read and interpret graphs on a coordinate plane.
Ordered Pairs: A coordinate pair is a set of two values that locate a point on a grid system. You need to understand how ordered pairs work and how to plot them on a coordinate plane.
Slope: Slope is the steepness of a line and how it changes along the line. You need to know how to calculate slope and what different slopes look like when graphed.
Y-Intercept: The y-intercept is the point where a line crosses the y-axis on a graph. You need to know how to find the y-intercept and how to use it to graph lines.
Graphing Equations: Graphing equations is the process of plotting points on a graph and connecting them with a line. You need to know how to graph an equation in slope-intercept form and standard form.
Slope-Intercept Form: Slope-intercept form is a way to express a linear equation in the form of y = mx + b, where m is the slope and b is the y-intercept. You need to know how to graph lines in slope-intercept form.
Standard Form: Standard form is a way to express a linear equation in the form of Ax + By = C. You need to know how to convert equations into standard form and how to graph lines in standard form.
Point-Slope Form: Point-slope form is a way to express a linear equation in the form of y – y1 = m(x – x1), where m is the slope and (x1, y1) is a point on the line. You need to know how to graph lines in point-slope form.
Parallel and Perpendicular Lines: Parallel lines have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other. You need to know how to identify parallel and perpendicular lines when graphing equations.
Systems of Equations: A system of equations is a set of two or more equations with two or more variables. You need to know how to solve systems of equations, both algebraically and graphically.
Intercepts: An x-intercept is the point where a line crosses the x-axis on a graph, and a y-intercept is the point where a line crosses the y-axis. You need to know how to find intercepts and how to use them to graph lines.
Applications: Graphing linear equations is used in many different everyday applications, such as calculating the slope of a ramp, finding the best route to take on a map, and determining the maximum profit a company can make. You need to understand how to apply graphing linear equations to real-world situations.
Slope-Intercept Form: Allows the equation of a line to be written as y=mx+b, where m is the slope and b is the y-intercept.
Point-Slope Form: Allows the equation of a line to be written as y-y1=m(x-x1), where m is the slope and (x1,y1) is a point on the line.
Standard Form: Allows the equation of a line to be written as Ax+By=C, where A, B, and C are constants and A and B are not both zero.
Intercepts Form: Allows the equation of a line to be written as x/a + y/b = 1 or y = -(b/a)*x + b, where a and b are the x- and y-intercepts, respectively.
Parallel and Perpendicular Lines: Equations of lines that are parallel have the same slope, while lines that are perpendicular have slopes that are negative reciprocals (or opposite and reciprocal).
Graphing by Plotting Points: The equation of a line is graphed by finding two or more points on the line and plotting them on a Cartesian plane.
Graphing Using Slope and Y-Intercept: The equation of a line is graphed by determining its slope and y-intercept and then using that information to plot the line on a Cartesian plane.
Graphing Using a Table of Values: The equation of a line is graphed by creating a table of x- and y-values that satisfy the equation and then plotting those points on a Cartesian plane.
Graphing Using Intercepts: The equation of a line is graphed by finding the x- and y-intercepts and then plotting those points on a Cartesian plane.
Graphing by Transformations: The equation of a line is graphed by using transformations such as translations, reflections, and dilations to manipulate a basic linear function.
"The order of the coordinates is significant."
"They are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in 'the x-coordinate'."
"The coordinates are taken to be real numbers in elementary mathematics."
"The coordinates... may be complex numbers or elements of a more abstract system such as a commutative ring."
"The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa."
"This is the basis of analytic geometry."
"Coordinates... uniquely determine the position of the points or other geometric elements on a manifold."
"A coordinate system... allows problems in geometry to be translated into problems about numbers and vice versa."
"A coordinate system is a system that uses... coordinates, to uniquely determine the position of the points or other geometric elements."
"They are sometimes identified... by a letter, as in 'the x-coordinate'."
"A coordinate system... determines the position of the points or other geometric elements on a manifold such as Euclidean space."
"The use of a coordinate system is common in elementary mathematics."
"The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa."
"The coordinates... may be complex numbers."
"They are sometimes identified by their position in an ordered tuple."
"The coordinates... may be... elements of a more abstract system such as a commutative ring."
"Coordinates... uniquely determine the position of the points or other geometric elements on a manifold."
"The order of the coordinates is significant."
"The use of a coordinate system is the basis of analytic geometry."