Graphs Study Guide

In coordinate geometry, the relationship between lines can be determined by their gradients. The gradient (or slope) of a straight line measures its steepness. For a line defined by the equation y = mx + c, 'm' represents the gradient, and 'c' represents the y-intercept. For two lines to be parallel, they must have the exact same gradient. This means that if line 1 has a gradient m_1 and line 2 has a gradient m_2, then for them to be parallel, m_1 = m_2. Parallel lines never intersect.

Perpendicular lines, on the other hand, intersect at a right angle (90 degrees). The relationship between the gradients of two perpendicular lines is that their product is -1. If line 1 has a gradient m_1 and line 2 has a gradient m_2, then for them to be perpendicular, m_1 * m_2 = -1. This also implies that m_2 = -1/m_1. So, the gradient of a perpendicular line is the negative reciprocal of the original line's gradient. For example, if a line has a gradient of 2, a line perpendicular to it will have a gradient of -1/2. If a line has a gradient of -3/4, a perpendicular line will have a gradient of 4/3. This concept is fundamental for solving problems involving geometric properties of shapes on a coordinate plane, such as proving a quadrilateral is a rectangle or finding the equation of an altitude of a triangle. These principles apply whether the lines are given in the form y = mx + c or in the ax + by = c form, requiring rearrangement to find 'm'.

In GCSE Mathematics, translations and reflections are fundamental transformations that alter the position or orientation of a shape without changing its size or form. For higher methods, understanding the effect of combining multiple transformations, especially reflections, is crucial, as is working with algebraic representations of these transformations. A translation shifts every point of a shape by the same distance in the same direction, defined by a column vector. For example, a translation by vector $\begin{pmatrix} a \\ b \end{pmatrix}$ moves a point $(x, y)$ to $(x+a, y+b)$. Reflections, on the other hand, flip a shape over a line of reflection, known as the mirror line. Common mirror lines include the axes ($x=0$, $y=0$), the lines $y=x$ and $y=-x$, and general horizontal or vertical lines ($y=k$, $x=k$). Higher methods often involve performing a sequence of two reflections and identifying the single equivalent transformation. For instance, reflecting a shape in the line $x=a$ followed by a reflection in the line $x=b$ is equivalent to a single translation. Similarly, reflecting in $y=a$ then $y=b$ is a translation. Reflecting in two intersecting lines is equivalent to a single rotation. The centre of rotation is the intersection of the two mirror lines, and the angle of rotation is twice the angle between the lines. Finally, understanding how the coordinates of points change under these transformations algebraically is vital. For example, reflecting $(x, y)$ in the line $y=x$ maps it to $(y, x)$, and reflecting in $y=-x$ maps it to $(-y, -x)$. These higher applications require a more abstract understanding beyond simple plotting.