Coordinate Geometry

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infoWhy this? Straight-line graphs make algebraic relationships visible and allow rates of change and starting values to be interpreted geometrically. Understanding gradient, intercepts, and parallel lines supports modelling and comparison of linear situations.

scheduleWhy now? Year 7 coordinate geometry established plotting and simple line equations, while algebra and equation work provide the necessary symbolic fluency. We can now generalise these ideas through y = mx + c before studying more complex linear and non-linear graphs.

A linear relationship can be modelled by a straight line on a coordinate grid.
A point on a graph is written as an ordered pair `(x, y)`, where `x` is the horizontal coordinate and `y` is the vertical coordinate.
A table of values for a linear graph lists matching `x` and `y` values that form points lying on one straight line.
The equation of many straight-line graphs can be written in the form `y = mx + c`, where `m` is the gradient and `c` is the `y`-intercept.
The gradient of a straight line is the rate of change of `y` as `x` increases and can be found using `\text{gradient} = \frac{\text{change in } y}{\text{change in } x}`.
A positive gradient means the line rises from left to right; a negative gradient means it falls from left to right.
A gradient of zero means the line is horizontal and has equation `y = c` for some constant `c`.
The `y`-intercept is the point where the line crosses the `y`-axis, and at this point `x = 0`.
In `y = mx + c`, the constant `c` is the `y`-value when `x = 0` (the `y`-intercept).
The `x`-intercept is the point where the line crosses the `x`-axis, and at this point `y = 0`.
Two non-vertical lines are parallel when they have the same gradient.
If two lines have different gradients, they are not parallel.
Vertical lines have equations of the form `x = a`.
Two different parallel lines have the same `m` value in `y = mx + c` but different `c` values.
The gradient can be interpreted as "for every 1 increase in `x`, `y` changes by `m`".
The gradient read from a graph is the same no matter which two points on the line are chosen (as long as they are accurate).
A line can be uniquely identified by its gradient and one other piece of information, such as its `y`-intercept or a point it passes through.
When plotting a graph, equal steps on an axis must represent equal numerical changes (a linear scale).

rocket_launchYou must be able to

Choose a sensible linear scale for both axes (using most of the grid and consistent intervals) and label axes with the correct variables and units if given.
Plot a linear graph from a table by marking each ordered pair `(x, y)` accurately and drawing a single straight line through the points with a ruler.
Calculate the gradient from a graph by selecting two well-spaced points on the line and computing `\text{gradient} = \frac{\text{change in } y}{\text{change in } x}` using a clear right-angled triangle (“rise over run”).
Identify the `y`-intercept on a graph by finding where the line crosses the `y`-axis and reading off the `y`-value accurately.
Interpret a gradient in context by describing how much `y` increases or decreases when `x` increases by 1 (or another chosen step).
Write the equation of a line from a graph by finding the gradient `m` and `y`-intercept `c`, then substituting into `y = mx + c`.
Write an equation for a line parallel to `y = mx + c` by keeping the same gradient `m` and using a new intercept `c` (or another given point) to determine the new equation.
Check whether two given straight-line equations represent parallel lines by comparing their gradient `m` values.

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