The coordinate plane, also known as the Cartesian coordinate system, is a two-dimensional plane used to represent and analyze the relationship between two variables. It consists of a horizontal x-axis and a vertical y-axis, which intersect at a point called the origin, forming a grid-like structure that allows for the precise location and graphing of points, lines, and other mathematical objects.
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The coordinate plane is used to graph and analyze linear equations, inequalities, systems of equations, and quadratic equations in two variables.
The x-axis represents the horizontal direction, while the y-axis represents the vertical direction, allowing for the precise location of points on the plane.
The origin, where the x-axis and y-axis intersect, is the point with coordinates (0, 0) and serves as the reference point for all other coordinates on the plane.
The four quadrants of the coordinate plane are defined by the positive or negative signs of the x and y coordinates, with each quadrant having a unique combination of signs.
The slope of a line on the coordinate plane can be determined by the ratio of the change in y-coordinates to the change in x-coordinates between any two points on the line.
Review Questions
Explain how the coordinate plane is used to graph linear equations in two variables.
The coordinate plane is essential for graphing linear equations in two variables, such as $y = mx + b$, where $m$ represents the slope of the line and $b$ represents the $y$-intercept. By plotting points using the $x$ and $y$ coordinates, the line can be drawn on the coordinate plane, allowing for the visualization and analysis of the relationship between the two variables. The slope of the line can be determined by the ratio of the change in $y$-coordinates to the change in $x$-coordinates between any two points on the line.
Describe how the coordinate plane is used to solve systems of linear equations.
The coordinate plane is a crucial tool for solving systems of linear equations graphically. By graphing the individual equations on the same coordinate plane, the point of intersection, if it exists, represents the solution to the system. The coordinates of the point of intersection give the values of the variables that satisfy both equations in the system. This graphical approach allows for the visualization of the relationship between the equations and can be used to determine the number of solutions, whether the system is consistent or inconsistent, and whether the lines are parallel, intersecting, or coincident.
Analyze how the coordinate plane is used to graph quadratic equations in two variables and describe the key features that can be identified.
When graphing quadratic equations in two variables, such as $y = ax^2 + bx + c$, the coordinate plane is used to plot the parabolic curve that represents the equation. The key features that can be identified on the coordinate plane include the vertex, which represents the minimum or maximum point of the parabolic curve, the $x$-intercepts, which are the points where the curve intersects the $x$-axis, and the $y$-intercept, which is the point where the curve intersects the $y$-axis. These features provide valuable information about the behavior and characteristics of the quadratic equation, allowing for a deeper understanding of its graphical representation and the relationship between the variables.