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neurologyYou need to know

• The Cartesian coordinate plane has four quadrants defined by the x-axis (horizontal) and y-axis (vertical), and points are written as ordered pairs `(x, y)`.
• In all four quadrants, positive x-values lie to the right, negative x-values to the left, positive y-values above the origin, and negative y-values below the origin.
• A horizontal line has equation `y = k` (for constant `k`) and gradient 0; it crosses the y-axis at `(0, k)`.
• A vertical line has equation `x = k` (for constant `k`) and its gradient is undefined; it crosses the x-axis at `(k, 0)`.
• The line `y = x` is a straight line through the origin with gradient 1; the line `y = -x` is a straight line through the origin with gradient −1.
• A linear function in slope–intercept form `y = mx + c` represents a straight line with gradient `m` and y-intercept `(0, c)`.
• The gradient of a straight line is the ratio “rise over run”: `gradient = Δy / Δx`; between two points `(x_1, y_1)` and `(x_2, y_2)` it is `(y_2 - y_1) / (x_2 - x_1)` when `x_2 ≠ x_1`.
• The x-intercept of a non-vertical line `y = mx + c` (with `m ≠ 0`) occurs at `x = -c/m`, where the graph crosses the x-axis.
• Parallel lines have equal gradients; if a line has equation `y = mx + c`, any line parallel to it has equation `y = mx + d` for some constant `d`.
• Rearranging a linear equation like `ax + by = c` (with `b ≠ 0`) into the form `y = mx + c` gives `y = (-a/b)x + (c/b)`.
• 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` with `a ≠ 0`, and its graph is a parabola that opens upwards if `a > 0` and downwards if `a < 0`.
• For `y = ax^2 + bx + c`, the axis of symmetry is `x = -b/(2a)` and the turning point (vertex) is at `(-b/(2a), f(-b/(2a)))`.
• The y-intercept of a quadratic `y = ax^2 + bx + c` is `(0, c)`; its roots (x-intercepts) are the x-values where `y = 0` and may be zero, one, or two real values.
• A cubic function can be written as `y = ax^3 + bx^2 + cx + d` with `a ≠ 0`; many cubics show an S-shaped curve and may have one or three real x-intercepts.
• A reciprocal function of the form `y = k/x` has asymptotes `x = 0` and `y = 0`; it has branches in quadrants I and III if `k > 0` and in quadrants II and IV if `k < 0`.
• An exponential function `y = a^x` with `a > 0` and `a ≠ 1` passes through `(0, 1)`; it shows growth if `a > 1` and decay if `0 < a < 1`.
• Plotting a graph from a table of values requires substituting chosen x-values into the function to compute corresponding y-values and plotting the resulting coordinates.
• A straight line can be drawn from its equation `y = mx + c` without a table by plotting the y-intercept `(0, c)` and using the gradient `m` to locate a second point.
• On any graph, intercepts and key points (such as roots and turning points) are read as coordinate pairs `(x, y)` to an appropriate degree of accuracy from the axes and grid.

• Plot coordinates accurately in all four quadrants, marking each point at the exact intersection of its x- and y-values and labelling key points clearly.
• Draw horizontal and vertical lines from their equations by identifying `y = k` or `x = k` and sketching a straight line through the corresponding intercept with the correct orientation.
• Generate a table of values for a linear equation by substituting at least three x-values, calculate the corresponding y-values, and plot a straight line through the points with a ruler.
• Estimate the gradient of a straight line graphically by constructing a right-angled triangle on the line, reading scaled Δx and Δy from the axes, and computing `gradient = Δy/Δx`.
• Read y- and x-intercepts from a plotted line by identifying where the graph crosses the axes and recording the exact or estimated coordinates.
• Write the equation of a line parallel to a given line by using the same gradient and determining the new y-intercept from a given point on the required line.
• Reduce linear equations to the form `y = mx + c` by rearranging algebraically (including when given in forms like `ax + by = c`), stating the gradient and y-intercept once simplified.
• Draw a straight line directly from `y = mx + c` by plotting the y-intercept and using the gradient to locate a second point (rise–run), then extending with a ruler.
• Find the midpoint of a line segment by substituting the endpoints into `((x_1 + x_2)/2, (y_1 + y_2)/2)` and writing the coordinate to the correct precision.
• Plot a quadratic graph from a table of values by choosing symmetric x-values around the axis of symmetry when possible, calculating y-values, plotting points, and drawing a smooth parabola through them without kinks or corners.