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neurologyYou need to know

• A straight-line graph has equation in gradient–intercept form `y = mx + c`, where `m` is the gradient (change in `y` per unit change in `x`) and `c` is the `y`-intercept.
• For a line in standard form `ax + by = c` with `b ≠ 0`, the gradient is `-a/b`, the `y`-intercept is `c/b`, and the `x`-intercept is `c/a` when `a ≠ 0`.
• Two non-vertical lines are perpendicular if and only if the product of their gradients is `-1`; equivalently, if a line has gradient `m`, any perpendicular line has gradient `-1/m`.
• The midpoint of a line segment joining `(x_1, y_1)` and `(x_2, y_2)` is `((x_1 + x_2)/2, (y_1 + y_2)/2)`.
• A quadratic function has the form `y = ax^2 + bx + c` (`a ≠ 0`); its roots are the `x`-coordinates where the graph crosses the `x`-axis (`y = 0`), and its `y`-intercept is `c` (the value at `x = 0`).
• The axis of symmetry of `y = ax^2 + bx + c` is the vertical line `x = -b/(2a)`; the turning point (vertex) lies on this line.
• Completing the square writes a quadratic as `y = a(x - h)^2 + k` (with `a ≠ 0`), so the turning point is `(h, k)` and the graph opens upwards if `a > 0` and downwards if `a < 0`.
• A cubic function typically has an S-shaped curve and may have one or three real `x`-intercepts; for `y = x^3` the graph is increasing for all `x` and is symmetric about the origin.
• A reciprocal graph `y = k/x` (`k ≠ 0`) has asymptotes `x = 0` and `y = 0`; it lies in opposite quadrants when `k > 0` and adjacent quadrants when `k < 0`.
• An exponential function `y = ab^x` with `a > 0` and `b > 0`, `b ≠ 1`, has horizontal asymptote `y = 0`, passes through `(0, a)`, increases if `b > 1` and decreases if `0 < b < 1`.
• The basic trigonometric graphs in degrees satisfy: `y = sin x` and `y = cos x` have amplitude 1 and period `360°`, with key points at multiples of `90°`, while `y = tan x` has period `180°` and vertical asymptotes at `x = 90° + 180°n`.
• The solutions to a pair of simultaneous equations correspond to the coordinates of the intersection points of their graphs.
• Graphical features are interpreted as follows: roots are points where `y = 0`, the `y`-intercept is where `x = 0`, and turning points are local maxima or minima where the curve changes direction.
• Linear inequalities in two variables define half-planes; a solid boundary line is used for `≤` or `≥`, a dashed boundary for `<` or `>`, and the correct region is selected by testing a point.
• A circle centred at the origin with radius `r` has equation `x^2 + y^2 = r^2`; any point `(x, y)` on the circle satisfies this equation.
• For the circle `x^2 + y^2 = r^2`, the radius to a point `(x_1, y_1)` on the circle is perpendicular to the tangent at that point, and the tangent’s equation can be written as `x x_1 + y y_1 = r^2`.
• The gradient–intercept method allows a straight line to be drawn using one point (often the intercept) and the gradient to locate a second point.
• A table of values enables accurate plotting of non-linear graphs; substituting chosen `x`-values into the function generates ordered pairs for plotting.
• The turning point of `y = ax^2 + bx + c` can be read off after completing the square, with `h = -b/(2a)` and `k = c - b^2/(4a)` when `a = 1` (and adjusted accordingly for general `a`).

• Rearrange a linear equation into `y = mx + c`, isolating `y` and simplifying to identify the gradient and intercept clearly.
• Draw a straight-line graph directly from its equation by plotting the `y`-intercept and using the gradient to locate a second point, then drawing a accurate line through them with a ruler.
• Find the midpoint of a line segment by substituting the endpoint coordinates into `((x_1 + x_2)/2, (y_1 + y_2)/2)` and stating the result as an ordered pair.
• Plot quadratic, cubic, reciprocal, and exponential graphs by constructing a table of values, calculating accurate coordinates, and marking any asymptotes where appropriate.
• Identify from a graph the roots (`x`-intercepts), the `y`-intercept, and the turning point(s), labelling their coordinates to an appropriate degree of accuracy.
• Solve a pair of simultaneous equations graphically by plotting both graphs on the same axes and reading the intersection point(s) with appropriate scale-based accuracy.
• Write the equation of a line perpendicular to a given line by determining its gradient and using the negative reciprocal with a given point in the point–gradient form `y - y_1 = m(x - x_1)`.
• Complete the square for `y = ax^2 + bx + c` to express it as `y = a(x - h)^2 + k`, then state the turning point `(h, k)` and the axis of symmetry `x = h`.
• Represent regions defined by linear inequalities by drawing the correct boundary lines (solid or dashed) and shading or labelling the appropriate half-plane after a test-point check.
• Use the circle equation `x^2 + y^2 = r^2` to find the radius from a point and form the tangent’s equation at `(x_1, y_1)` via `x x_1 + y y_1 = r^2`, or by using the perpendicular gradient and a point on the tangent.