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Questions and Answers
To plot a point (x, y), you start at the origin and move x units up along the y-axis and y units right along the x-axis.
To plot a point (x, y), you start at the origin and move x units up along the y-axis and y units right along the x-axis.
False
The origin in a 2D Cartesian coordinate system is represented by the point (1, 1).
The origin in a 2D Cartesian coordinate system is represented by the point (1, 1).
False
The distance between two points (x1, y1) and (x2, y2) is given by the formula: |x2 - x1| + |y2 - y1|.
The distance between two points (x1, y1) and (x2, y2) is given by the formula: |x2 - x1| + |y2 - y1|.
False
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula: ((x2 - x1)/2, (y2 - y1)/2).
The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the formula: ((x2 - x1)/2, (y2 - y1)/2).
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The quadrants in a 2D Cartesian coordinate system are labeled A, B, C, and D.
The quadrants in a 2D Cartesian coordinate system are labeled A, B, C, and D.
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The x-axis is the vertical axis in a 2D Cartesian coordinate system.
The x-axis is the vertical axis in a 2D Cartesian coordinate system.
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In a 2D Cartesian coordinate system, the point (0, 0) is not part of any quadrant.
In a 2D Cartesian coordinate system, the point (0, 0) is not part of any quadrant.
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The x-axis is the line where the y-coordinate is always zero.
The x-axis is the line where the y-coordinate is always zero.
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A point in the fourth quadrant has a positive x-coordinate and a negative y-coordinate.
A point in the fourth quadrant has a positive x-coordinate and a negative y-coordinate.
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The coordinates of a point in the second quadrant are always in the form (x, y) where x > 0 and y > 0.
The coordinates of a point in the second quadrant are always in the form (x, y) where x > 0 and y > 0.
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The graph of a point (x, y) is always to the right of the y-axis if x > 0.
The graph of a point (x, y) is always to the right of the y-axis if x > 0.
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The y-coordinate of a point is always positive in the first and second quadrants.
The y-coordinate of a point is always positive in the first and second quadrants.
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The x-axis and y-axis are perpendicular to each other in a 2D Cartesian coordinate system.
The x-axis and y-axis are perpendicular to each other in a 2D Cartesian coordinate system.
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The coordinates of the origin are always (1, 1) in a 2D Cartesian coordinate system.
The coordinates of the origin are always (1, 1) in a 2D Cartesian coordinate system.
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Study Notes
Cartesian Coordinates
Definition
- A way of representing points in space using ordered pairs of numbers (x, y) or (x, y, z) in 2D or 3D space, respectively.
- Also known as rectangular coordinates.
Key Concepts
- X-axis: The horizontal axis, where x-coordinates are measured.
- Y-axis: The vertical axis, where y-coordinates are measured.
- Origin: The point where the x-axis and y-axis intersect, represented by (0, 0).
- Quadrants: The four regions formed by the x-axis and y-axis, labeled I, II, III, and IV.
Plotting Points
- To plot a point, start at the origin and move:
- Right x units along the x-axis.
- Up y units along the y-axis.
- Example: Plot the point (3, 4).
- Move 3 units right along the x-axis.
- Move 4 units up along the y-axis.
Distance Formula
- The distance between two points (x1, y1) and (x2, y2) is given by:
- √((x2 - x1)^2 + (y2 - y1)^2)
Midpoint Formula
- The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by:
- ((x1 + x2)/2, (y1 + y2)/2)
Cartesian Coordinates
Definition
- A system of representing points in space using ordered pairs of numbers (x, y) or (x, y, z) in 2D or 3D space.
Key Concepts
- X-axis: The horizontal axis where x-coordinates are measured.
- Y-axis: The vertical axis where y-coordinates are measured.
- Origin: The point where the x-axis and y-axis intersect, represented by (0, 0).
- Quadrants: The four regions formed by the x-axis and y-axis, labeled I, II, III, and IV.
Plotting Points
- To plot a point, start at the origin and move right x units along the x-axis and up y units along the y-axis.
- Example: Plot the point (3, 4) by moving 3 units right along the x-axis and 4 units up along the y-axis.
Distance Formula
- The distance between two points (x1, y1) and (x2, y2) is given by the square root of the sum of the squares of the differences in x and y coordinates.
Midpoint Formula
- The midpoint of a line segment with endpoints (x1, y1) and (x2, y2) is given by the average of the x and y coordinates of the two endpoints.
Cartesian Coordinates
Definition
- A system for assigning unique coordinates to points in a 2D plane using two perpendicular lines (x-axis and y-axis)
- Also known as rectangular coordinates
Key Components
- X-axis: horizontal line with positive values to the right and negative values to the left
- Y-axis: vertical line with positive values up and negative values down
- Origin: point where the x-axis and y-axis intersect, represented by (0, 0)
Coordinates of a Point
- A point is represented by an ordered pair of real numbers (x, y)
- x-coordinate: horizontal distance from the y-axis to the point
- y-coordinate: vertical distance from the x-axis to the point
Quadrants
Quadrant Division
- The coordinate plane is divided into four quadrants:
Quadrant I
- x > 0, y > 0 (upper-right)
Quadrant II
- x < 0, y > 0 (upper-left)
Quadrant III
- x < 0, y < 0 (lower-left)
Quadrant IV
- x > 0, y < 0 (lower-right)
Graphing Points
- To graph a point, start at the origin and move:
- x units to the right (if x > 0) or left (if x < 0)
- y units up (if y > 0) or down (if y < 0)
Importance of Cartesian Coordinates
- Used in physics, engineering, and computer science
- Enables representation of geometric shapes, functions, and equations in a 2D plane
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Description
Test your understanding of Cartesian coordinates, including x and y axes, origin, and quadrants. Learn how to represent points in 2D and 3D space.
