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neurologyYou need to know

• A vertical line has equation `x = a` and has undefined gradient; it crosses the x-axis at `x = a`.
• A horizontal line has equation `y = b` and has gradient 0; it crosses the y-axis at `y = b`.
• The graph of `y = x` is a straight line through the origin with gradient 1.
• The graph of `y = -x` is a straight line through the origin with gradient `-1`.
• In `y = mx + c`, `m` is the gradient (rate of change) and `c` is the y-intercept where the line crosses the y-axis.
• Two non-vertical lines are parallel if and only if they have equal gradients.
• Two non-vertical lines are perpendicular if and only if the product of their gradients is `-1` (i.e., `m_1 m_2 = -1`).
• Any linear equation in two variables can be rearranged into the form `y = mx + c` using inverse operations.
• The gradient of the line through points `(x_1, y_1)` and `(x_2, y_2)` is `m = (y_2 - y_1) / (x_2 - x_1)`.
• The midpoint of the line segment joining `(x_1, y_1)` and `(x_2, y_2)` is `((x_1 + x_2)/2, (y_1 + y_2)/2)`.
• The general linear form `ax + by = c` represents a straight line provided not both `a` and `b` are zero.
• A quadratic function `y = ax^2 + bx + c` graphs as a parabola opening upwards if `a > 0` and downwards if `a < 0`.
• The x-intercepts (roots) of a graph are the values of `x` where `y = 0`, and the y-intercept is the value of `y` when `x = 0`.
• The turning point of `y = ax^2 + bx + c` occurs at `x = -b/(2a)`, and in completed square form `y = a(x - h)^2 + k` the turning point is `(h, k)`.
• A typical cubic function has an S-shaped curve when the leading coefficient is positive and crosses the y-axis at `y = f(0)`.
• The reciprocal function `y = k/x` has asymptotes `x = 0` and `y = 0`, lying in quadrants I and III when `k > 0` and in II and IV when `k < 0`.
• An exponential function `y = a^x` with `a > 1` passes through `(0, 1)` and is increasing; with `0 < a < 1` it is decreasing.
• The sine and cosine graphs `y = \,\sin x` and `y = \,\cos x` (degrees) have amplitude 1 and period `360°`, while `y = \,\tan x` has period `180°` with vertical asymptotes at `x = 90° + 180°n` for integer `n`.
• The solution(s) to a pair of simultaneous equations correspond to the coordinate(s) of the intersection point(s) of their graphs.
• Key features of non-linear graphs that can be read directly include roots (where `y = 0`), intercepts, turning points, and any asymptotes.

• Plot vertical lines `x = a`, horizontal lines `y = b`, and the lines `y = x` and `y = -x` accurately on Cartesian axes.
• Calculate the gradient of a straight line from a graph by selecting two clear points and computing `m = rise/run` exactly or to a stated accuracy.
• Identify the y-intercept from a graph and write the line’s equation in the form `y = mx + c` by combining gradient and intercept.
• Rearrange linear equations (including `ax + by = c`) into `y = mx + c` using inverse operations, showing clear algebraic steps.
• Draw a linear graph directly from `y = mx + c` by marking the y-intercept `c` and using the gradient `m` to locate a second point before drawing the line with a ruler.
• Write the equation of a line parallel or perpendicular to a given line through a specified point, ensuring equal gradient for parallel and `m_1 m_2 = -1` for perpendicular.
• Find the midpoint of a line segment by substituting the endpoints into `((x_1 + x_2)/2, (y_1 + y_2)/2)` and plotting it correctly.
• Plot quadratic, cubic, reciprocal, and exponential graphs from a table of values using an appropriate scale and a smooth curve through the plotted points.
• Read off roots, intercepts, and turning points from a plotted graph to a specified degree of accuracy and state them with correct coordinates.
• Solve simultaneous linear equations graphically by drawing both lines on the same axes and stating the intersection point as the solution; check by substitution.