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neurologyYou need to know

• A linear equation in the form `y=mx+c` has a graph that is a straight line.
• In `y=mx+c`, `m` is the gradient (slope) and `c` is the y-intercept, the value of `y` when `x=0`.
• Gradient is the change in `y` divided by the change in `x`, so `m=\Delta y/\Delta x` and can be interpreted as “rise over run”.
• A positive gradient slopes upwards from left to right, and a negative gradient slopes downwards from left to right.
• The y-intercept is the point where a graph crosses the y-axis and has coordinates `(0,c)` for a line in the form `y=mx+c`.
• The x-intercept is the point where a graph crosses the x-axis and always has `y=0`.
• The roots (or zeros) of a graph are the x-values where `y=0`, so each root is the x-coordinate of an x-intercept.
• Rearranging a linear equation means using inverse operations to make `y` the subject, ending in the form `y=mx+c`.
• A line segment joining points `A(x_1,y_1)` and `B(x_2,y_2)` has a midpoint exactly halfway between them.
• The midpoint of `A(x_1,y_1)` and `B(x_2,y_2)` is `\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)`.
• A quadratic function can be written as `y=ax^2+bx+c`, and its graph is a parabola that opens upwards or downwards.
• For a quadratic, if `a>0` the parabola opens upwards (has a minimum), and if `a<0` it opens downwards (has a maximum).
• Quadratic graphs are smooth curves and should not be drawn by joining points with straight line segments.
• A table of values for a function lists chosen x-values and the corresponding y-values found by substitution.
• A turning point (vertex) on a quadratic graph is the point where the graph changes direction and has either the lowest or highest y-value on the curve.
• The y-intercept of any graph is found where it crosses the y-axis (at `x=0`), and for a quadratic `y=ax^2+bx+c` this is `(0,c)`.
• Graphical features can be estimated by reading coordinates from the grid, so answers are usually approximate unless the point lies exactly on grid intersections.

• Rearrange a given linear equation into the form `y=mx+c` by collecting like terms and using inverse operations so that `y` is alone on one side.
• Draw a straight-line graph from `y=mx+c` without a table by plotting the y-intercept `(0,c)` and using the gradient `m` as a rise/run to plot a second point, then drawing a straight line through the points.
• Convert a gradient given as a fraction (for example `m=\frac{3}{2}`) into a repeatable step pattern on the grid (up 3, right 2) to plot points accurately.
• Find the midpoint of a line segment between `A(x_1,y_1)` and `B(x_2,y_2)` by averaging the x-coordinates and averaging the y-coordinates.
• Construct a table of values for a quadratic by substituting each chosen x-value into the equation and calculating the corresponding y-value accurately.
• Plot a quadratic graph by marking the coordinate pairs from the table and sketching a smooth, symmetric curve through them.
• Recognise whether a given graph represents a quadratic by identifying a parabola shape with a single turning point (opening up or down).
• Find roots (x-intercepts), the y-intercept, and the turning point from a graph by reading the relevant coordinates from the axes and the vertex, stating answers to an appropriate degree of accuracy.