The coordinate plane and graphs
The coordinate plane turns equations into pictures. Plotting points, finding distances and midpoints, and locating intercepts are the first steps in reading any graph.
The coordinate plane
Two perpendicular number lines, the horizontal x-axis and vertical y-axis, cross at the origin (0, 0). Every point is an ordered pair (x, y): move x units horizontally, then y units vertically. The axes divide the plane into four quadrants, numbered I to IV anticlockwise from the upper right, where both coordinates are positive. Rene Descartes popularised this system in 1637.
Graphs of equations
The graph of an equation in x and y is the set of all points whose coordinates make it true. For \(y = 2x - 1\), the points (0, -1), (1, 1), and (2, 3) all lie on the graph, which is a straight line. Plotting several points from a table of values reveals a graph's shape.
Intercepts
The x-intercept is where a graph crosses the x-axis (y = 0); the y-intercept is where it crosses the y-axis (x = 0). For \(3x + 2y = 12\): setting y = 0 gives x = 4, so (4, 0); setting x = 0 gives y = 6, so (0, 6). Two intercepts are enough to draw a line.
The distance formula
The distance between \((x_1, y_1)\) and \((x_2, y_2)\) is \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), which is Pythagoras's theorem applied to the horizontal and vertical gaps. From (1, 2) to (7, 10), the gaps are 6 and 8, so the distance is \(\sqrt{36 + 64} = 10\).
The midpoint formula
The midpoint of the segment between two points averages their coordinates: \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\). The midpoint of (1, 2) and (7, 10) is (4, 6).
Circles
A circle is the set of points at a fixed distance r from a centre (h, k). The distance formula gives its equation: \((x - h)^2 + (y - k)^2 = r^2\). So \((x - 2)^2 + (y + 1)^2 = 25\) is a circle with centre (2, -1) and radius 5.
Symmetry
Graphs can be symmetric about the y-axis (if (x, y) is on the graph, so is (-x, y), as for y = x squared), about the x-axis, or about the origin (if (x, y) is on it, so is (-x, -y), as for y = x cubed). Spotting symmetry halves the work of plotting.
Worked example
A delivery drone flies from a depot at (0, 0) to a house at (3, 4), with units in kilometres. The straight-line distance is the square root of (9 + 16) = 5 km. A charging pad halfway along the route would be at the midpoint, (1.5, 2).
Key takeaways
Points are ordered pairs (x, y) located from the origin, and the axes create four quadrants. A graph contains every point satisfying its equation. Find intercepts by setting y = 0 for the x-intercept and x = 0 for the y-intercept. The distance formula comes from Pythagoras's theorem, and the midpoint averages coordinates. A circle with centre (h, k) and radius r has equation (x - h) squared + (y - k) squared = r squared. Symmetry about an axis or the origin simplifies graphing.
Flashcard
(0, 0)
Check your understanding
What is the distance between (1, 2) and (7, 10)?