The explanation
Understanding the Cartesian Plane and Coordinates
The Cartesian plane is a two-dimensional surface formed by the intersection of two perpendicular number lines called axes. The horizontal number line is called the x-axis, and the vertical number line is called the y-axis. The single point where these two axes cross at right angles is known as the origin, denoted by the coordinates (0, 0). The two axes divide the plane into four regions called quadrants, which are numbered counter-clockwise using Roman numerals from one to four, starting from the top-right quadrant where both coordinates are positive.
Every point on the Cartesian plane is identified by an ordered pair of numbers enclosed in parentheses, written as (x, y). The first number, x, is the x-coordinate, representing the horizontal distance and direction from the origin. A positive x-value moves to the right, and a negative x-value moves to the left. The second number, y, is the y-coordinate, representing the vertical distance and direction from the origin. A positive y-value moves upward, and a negative y-value moves downward. When plotting a point, movement always begins at the origin, travelling horizontally along the x-axis first, followed by a vertical movement parallel to the y-axis.
In the first quadrant, both the x-coordinate and the y-coordinate are positive, written as (+, +). In the second quadrant, located in the top-left, the x-coordinate is negative and the y-coordinate is positive, written as (-, +). In the third quadrant, located in the bottom-left, both coordinates are negative, written as (-, -). In the fourth quadrant, located in the bottom-right, the x-coordinate is positive and the y-coordinate is negative, written as (+, -). Recognizing the sign conventions for each quadrant allows for rapid verification when plotting coordinates and identifying errors in spatial reasoning.
Constructing a Table of Values for Linear Equations
A linear equation in two variables describes a relationship between an x-value and a y-value such that when all valid pairs are plotted on a Cartesian plane, they form a straight line. The general form of a linear equation is y equals mx plus c, where m represents the gradient or steepness of the line, and c represents the y-intercept, which is the point where the line crosses the vertical y-axis. To visualise this relationship graphically, a table of values is constructed.
A table of values is a grid that pairs chosen x-values with their corresponding calculated y-values. Selecting a set of consecutive integer x-values, such as from negative two to positive two, provides a balanced range that spans both sides of the y-axis. To find the y-value for a chosen x-value, substitute the number into the algebraic expression on the right side of the equation and evaluate the result using standard order of operations.
For example, given the equation y = 2x + 1, choosing an x-value of 3 requires multiplying 3 by 2 to get 6, and then adding 1 to obtain a y-value of 7. This forms the coordinate pair (3, 7). Calculating at least three or four distinct points ensures accuracy, as a fourth point acts as a check to confirm all calculated points lie on the same straight line. Once the table is complete, each column represents a single coordinate pair ready for plotting.
Plotting Linear Graphs
Plotting a linear graph involves transferring the coordinate pairs from a table of values onto a Cartesian grid. The process begins by drawing and labelling the x-axis and y-axis with appropriate, consistent numerical scales that accommodate the highest and lowest values present in the table. Each axis requires arrows at both ends to indicate that the number lines extend indefinitely in both directions, along with evenly spaced tick marks labelled with integer values.
To plot each point, locate the x-coordinate on the horizontal axis, move vertically to the corresponding y-coordinate value, and mark a clear cross or dot at the intersection. After plotting all calculated points, a ruler is aligned across the marks. If the calculations are correct, every plotted point will lie precisely on a single straight line. A pencil line is then drawn through all the points, extending past the outermost points to the edges of the grid, and labelled with the equation of the line.
Drawing a straight line through discrete points implies that the relationship holds true for all intermediate values, including decimals and fractions between the chosen integers. Extending the line beyond the plotted points demonstrates that the linear relationship is infinite. If any plotted point does not align with the others, it indicates an arithmetic error in the table of values, requiring a recalculation of that specific coordinate pair before drawing the final line.
Worked Examples of Plotting Linear Graphs
The process of generating a table of values and plotting a straight-line graph is demonstrated through systematic worked examples. These examples cover different types of linear equations, including those with positive coefficients, negative coefficients, and constant terms involving subtraction.
For the first example, consider the equation y = x + 2. A table of values is constructed by choosing x-values of -2, -1, 0, 1, and 2. Substituting these into the equation yields y-values of 0, 1, 2, 3, and 4 respectively. The resulting coordinate pairs are (-2, 0), (-1, 1), (0, 2), (1, 3), and (2, 4). Plotting these points on the Cartesian plane and joining them with a straight line produces a graph that slants upwards from left to right, crossing the y-axis at 2.
For the second example, consider the equation y = 2x - 3. Choosing x-values of -1, 0, 1, 2, and 3, the calculations are as follows: for x = -1, y = 2(-1) - 3 = -5; for x = 0, y = 2(0) - 3 = -3; for x = 1, y = 2(1) - 3 = -1; for x = 2, y = 2(2) - 3 = 1; for x = 3, y = 2(3) - 3 = 3. The coordinate pairs are (-1, -5), (0, -3), (1, -1), (2, 1), and (3, 3). Plotting these points creates a steeper upward-sloping line that crosses the y-axis at -3.
For the third example, consider an equation with a negative gradient, such as y = -x + 4. Choosing x-values of 0, 1, 2, 3, and 4 yields the following y-values: for x = 0, y = 4; for x = 1, y = 3; for x = 2, y = 2; for x = 3, y = 1; for x = 4, y = 0. The coordinate pairs are (0, 4), (1, 3), (2, 2), (3, 1), and (4, 0). Plotting these points produces a line that slants downwards from left to right, demonstrating that negative coefficients create decreasing linear relationships.
