Coordinates in Four Quadrants
Coordinates in four quadrants refer to the system used to locate points on a plane using two perpendicular lines, the x-axis and y-axis. The four quadrants are labeled I, II, III, and IV, and each quadrant represents a different combination of positive and negative x and y values. This system is commonly used in graphing and geometry to pinpoint exact locations.
3 Key excerpts on "Coordinates in Four Quadrants"
- Caryl Lorandini(Author)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...The point of intersection of the axes is called the origin and has coordinates (0, 0). Coordinates An ordered pair of numbers that identifies the location of a point on a coordinate plane, written as (x, y). Origin The point on the coordinate plane where the x - and y -axes intersect; has coordinates (0, 0). Quadrant One of four sections of a coordinate grid separated by horizontal and vertical axes; they are numbered I, II, III, and IV, counterclockwise from the upper right. x-axis The horizontal axis; the line whose equation is y = 0. x-intercept The point where a graph of an equation crosses the x -axis when (y = 0). y-axis The vertical axis; the line whose equation is x = 0. y-intercept The point where a graph of an equation crosses the y -axis when (x = 0). 12.1 What Is Coordinate Geometry? The x -axis runs horizontally, which is left and right. The y -axis is perpendicular to the x -axis and runs vertically, which is up and down. The axes divide the coordinate plane into four equal regions called quadrants. They are numbered with Roman numerals by starting at (+, +) the top-right quadrant and moving in a counterclockwise direction. The points on the plane are found by using ordered pairs of (x, y). The x always comes before the y, just like in the alphabet. Remember, the intersection of two lines is called a point. You can use lines or your finger if it helps. Think, “Find the elevator,” and then go up or down. The point (2, 3) is located by moving from the origin 2 to the right and 3 up. An ordered pair gives direction and distance. The direction is from the sign and the distance is the number. EXAMPLE 12.1 1) Graph and label each of the following points on the coordinate plane...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...4 Analytic Geometry 1. Rectangular Coordinates The points in a plane may be placed in one-to-one correspondence with pairs of real numbers. A common method is to use perpendicular lines that are horizontal and vertical and intersect at a point called the origin. These two lines constitute the coordinate axes; the horizontal line is the x-axis and the vertical line is the y-axis. The positive direction of the x-axis is to the right whereas the positive direction of the y-axis is up. If P is a point in the plane one may draw lines through it that are perpendicular to the x- and y-axes (such as the broken lines of Figure 4.1). The lines intersect the x-axis at a point with coordinate x 1 and the y-axis at a point with coordinate y 1. We call x 1 the x-coordinate or abscissa and y 1 is termed the y-coordinate or ordinate of the point P. Thus, point P is associated with the pair of real numbers (x 1, y 1) and is denoted P (x 1, y 1). The coordinate axes divide the plane into quadrants I, II, III, and IV. FIGURE 4.1. Rectangular coordinates. 2. Distance between Two Points; Slope The distance d between the two points P 1 (x 1, y 1) and P 2 (x 2, y 2) is d = (x 2 − x 1) 2 + (y 2 − y 1) 2 In the special case when P 1 and P 2 are both on one of the coordinate axes, for instance, the x-axis, d = (x 2 − x 1) 2 = | x 2 − x 1 |, or on the. y-axis, d = (y 2 − y 1) 2 = | y 2 − y 1 |. The midpoint of the line segment P 1 P 2 is (x 1 + x 2 2, y 1 + y 2 2). The slope of the line segment P 1 P 2, provided it is not vertical, is denoted by m and is given by m = y 2 − y 1 x 2 − x 1. The slope is related to the angle of inclination α (Figure 4.2) by m = tan α Two lines (or line segments) with...
eBook - ePub- Joy Ko, Kyle Steinfeld(Authors)
- 2018(Publication Date)
- Routledge(Publisher)
...Before examining these other representations in three-dimensional space, let’s begin with a non-rectangular coordinate system in the two-dimensional plane: polar coordinates. An alternate route to locating a position on the plane is to use two values: a distance from a reference point, and an angular displacement from a reference direction. As portrayed in the nearby diagram, these two values, denoted by the pair (r, θ), is related to the rectangular coordinates (x,y) in ℝR 2 using the formulas: Here, r is the distance from the origin and θ is the angle swept counterclockwise from the x-axis. A caveat of using polar coordinates is that they are inherently non-unique. We can see that (r,θ + 2πn) translates to the same point as (r,π). Further, there is the allowance for negative values of r, which produces additional opportunities for non-unique coordinates, such as (- r,π) = (r,π + r). In many computational applications, it is important to have a unique mapping between one set of coordinates to another. This may be accomplished by restricting the range of the given coordinates, limiting input values to fall within r > 0 and 0 # i 1 2r, for example. In summary, the polar coordinate system is defined by an origin point and an axis (the reference direction starting at the origin), and is evaluated by a radial coordinate r and an angular coordinate θ. The two most widely-used alternate coordinate systems in three-dimensional space are based on generalizations of polar coordinates to three dimensions. We present each in the section to follow, alongside their implementations as methods of the Decod.es CS class. fig 1.099 REPRESENTING A POINT IN POLAR COORDINATES Cylindrical Coordinates One generalization of polar coordinates to three-dimensions is the simple extrusion of a polar point along the z-axis...
