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Introduction to Graphs - The Cartesian Plane and Coordinates

Grade 8ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Cartesian Plane: A two-dimensional coordinate system formed by the intersection of two perpendicular number lines. The horizontal line is called the xx-axis and the vertical line is called the yy-axis. The point where they intersect is known as the Origin, denoted by O(0,0)O(0, 0). Visually, this creates a grid-like structure used to locate any position precisely.

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Coordinate Axes: The horizontal xx-axis and vertical yy-axis act as reference lines. The distance of a point from the yy-axis is its xx-coordinate (Abscissa), and its distance from the xx-axis is its yy-coordinate (Ordinate). Together, they are written as an ordered pair (x,y)(x, y).

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The Four Quadrants: The axes divide the plane into four regions called quadrants, numbered I to IV in a counter-clockwise direction. Quadrant I (top-right) contains points with (+,+)(+, +) signs; Quadrant II (top-left) contains (−,+)(-, +); Quadrant III (bottom-left) contains (−,−)(-,-); and Quadrant IV (bottom-right) contains (+,−)(+, -).

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Points on the Axes: Not all points lie within a quadrant. If a point lies exactly on the xx-axis, its yy-coordinate is always 00, represented as (x,0)(x, 0). If a point lies on the yy-axis, its xx-coordinate is always 00, represented as (0,y)(0, y). For example, (5,0)(5, 0) is on the xx-axis and (0,−3)(0, -3) is on the yy-axis.

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Plotting a Point: To plot a point P(a,b)P(a, b), start at the origin (0,0)(0, 0). Move 'aa' units along the xx-axis (right if positive, left if negative). From that position, move 'bb' units parallel to the yy-axis (up if positive, down if negative). Mark the final location with a dot.

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Linear Graphs: When points that satisfy a linear equation (like y=x+2y = x + 2) are plotted and joined, they form a straight line. Every point on this line represents a solution to the given equation.

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Independent and Dependent Variables: In a graph representing a real-world situation (like Time vs. Distance), the independent variable is usually plotted on the horizontal xx-axis, and the dependent variable is plotted on the vertical yy-axis.

📐Formulae

General representation of a point: P(x,y)P(x, y)

Coordinates of the Origin: O(0,0)O(0, 0)

Equation of the xx-axis: y=0y = 0

Equation of the yy-axis: x=0x = 0

General form of a linear equation: y=mx+cy = mx + c

💡Examples

Problem 1:

Identify the quadrant or axis for the following points without plotting them: A(3,4)A(3, 4), B(−2,5)B(-2, 5), C(0,−3)C(0, -3), and D(4,−2)D(4, -2).

Solution:

  1. For A(3,4)A(3, 4): Both xx and yy are positive (+,+)(+, +), so it lies in Quadrant I.
  2. For B(−2,5)B(-2, 5): xx is negative and yy is positive (−,+)(-, +), so it lies in Quadrant II.
  3. For C(0,−3)C(0, -3): The xx-coordinate is 00. Any point with x=0x=0 lies on the yy-axis.
  4. For D(4,−2)D(4, -2): xx is positive and yy is negative (+,−)(+, -), so it lies in Quadrant IV.

Explanation:

The location of a point is determined by the signs of its coordinates. If one coordinate is zero, the point lies on an axis rather than in a quadrant.

Problem 2:

Given the equation y=2x−1y = 2x - 1, find the coordinates of the points where the line crosses the xx-axis and the yy-axis.

Solution:

  1. To find the yy-axis intersection (where x=0x = 0): y=2(0)−1y = 2(0) - 1 y=−1y = -1 So, the point is (0,−1)(0, -1).

  2. To find the xx-axis intersection (where y=0y = 0): 0=2x−10 = 2x - 1 2x=12x = 1 x=frac12=0.5x = \\frac{1}{2} = 0.5 So, the point is (0.5,0)(0.5, 0).

Explanation:

Intersection with the yy-axis always occurs when x=0x=0, and intersection with the xx-axis always occurs when y=0y=0. We substitute these values into the linear equation to solve for the unknown coordinate.