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Algebra - Representing Relationships on the Cartesian Plane

Grade 7IB

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

🔑Concepts

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The Cartesian Plane is a two-dimensional surface formed by the intersection of two perpendicular number lines: the horizontal xx-axis and the vertical yy-axis. The point where they meet is called the origin, denoted as (0,0)(0,0). Visually, the axes divide the plane into four sections called Quadrants, labeled I (+,+)(+,+), II (−,+)(-,+), III (−,−)(-,-), and IV (+,−)(+,-) in a counter-clockwise direction.

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Ordered Pairs are written as (x,y)(x, y) and represent a specific location on the plane. The first value, the xx-coordinate, tells you how far to move left or right from the origin. The second value, the yy-coordinate, tells you how far to move up or down. For example, to plot (2,−3)(2, -3), you move 22 units right and 33 units down.

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A Linear Relationship is a mathematical connection between two variables that forms a straight line when plotted on a Cartesian plane. In these relationships, for every constant change in xx, there is a constant change in yy. Visually, this creates a perfectly straight path that does not curve.

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A Table of Values is a tool used to organize pairs of numbers that satisfy a specific algebraic rule. By choosing 'input' values for xx and calculating the corresponding 'output' values for yy, you create a list of coordinates to plot. For instance, for the rule y=x+2y = x + 2, the table would include pairs like (0,2)(0, 2), (1,3)(1, 3), and (2,4)(2, 4).

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The Gradient (or Slope) describes the steepness and direction of a line. It is the ratio of the vertical change (the 'rise') to the horizontal change (the 'run'). Visually, a positive gradient means the line goes 'uphill' from left to right, while a negative gradient means it goes 'downhill'. A horizontal line has a gradient of 00.

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Intercepts are points where the line crosses the axes. The yy-intercept is the point (0,c)(0, c) where the line crosses the vertical yy-axis; at this point, x=0x = 0. The xx-intercept is the point where the line crosses the horizontal xx-axis; at this point, y=0y = 0.

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Independent and Dependent Variables are used to describe the relationship between coordinates. The independent variable is usually represented on the xx-axis (e.g., time), while the dependent variable is represented on the yy-axis (e.g., distance). The value of yy 'depends' on what happens to xx.

📐Formulae

General Linear Equation (Slope-Intercept Form): y=mx+cy = mx + c

Gradient (Slope) Formula: m=riserun=y2−y1x2−x1m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}

Equation of a Horizontal Line: y=ky = k (where kk is a constant)

Equation of a Vertical Line: x=hx = h (where hh is a constant)

The Origin: (0,0)(0, 0)

💡Examples

Problem 1:

Create a table of values and plot the linear relationship given by the equation y=2x−1y = 2x - 1 for xx values of −1,0,1,2-1, 0, 1, 2.

Solution:

  1. Substitute each xx value into the equation:
  • For x=−1x = -1: y=2(−1)−1=−2−1=−3y = 2(-1) - 1 = -2 - 1 = -3. Point: (−1,−3)(-1, -3)
  • For x=0x = 0: y=2(0)−1=0−1=−1y = 2(0) - 1 = 0 - 1 = -1. Point: (0,−1)(0, -1)
  • For x=1x = 1: y=2(1)−1=2−1=1y = 2(1) - 1 = 2 - 1 = 1. Point: (1,1)(1, 1)
  • For x=2x = 2: y=2(2)−1=4−1=3y = 2(2) - 1 = 4 - 1 = 3. Point: (2,3)(2, 3)
  1. List the coordinates: (−1,−3),(0,−1),(1,1),(2,3)(-1, -3), (0, -1), (1, 1), (2, 3).
  2. Plot these points on the Cartesian plane and draw a straight line through them.

Explanation:

This approach uses substitution to find specific points. The resulting graph is a straight line with a yy-intercept at −1-1 and a gradient of 22 (meaning for every 11 unit moved right, the line goes up 22 units).

Problem 2:

Find the gradient (mm) of the line that passes through the points A(1,4)A(1, 4) and B(3,10)B(3, 10).

Solution:

  1. Identify the coordinates: (x1,y1)=(1,4)(x_1, y_1) = (1, 4) and (x2,y2)=(3,10)(x_2, y_2) = (3, 10).
  2. Use the gradient formula: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  3. Substitute the values: m=10−43−1m = \frac{10 - 4}{3 - 1}.
  4. Simplify the fraction: m=62=3m = \frac{6}{2} = 3.

Explanation:

The gradient is 33, which means the line is relatively steep and slopes upwards. In a real-world context, this would mean the dependent variable increases by 33 units for every 11 unit increase in the independent variable.