KS3MAT8 · Coordinates and graphs | KS3 Maths

Revise the key ideas

Cartesian coordinates — Write (x, y): horizontal first, vertical second. Axes cross at (0, 0). Negative values extend left and down, giving four quadrants. Use equal intervals on each individual axis, though the two axes may use different scales.
Plotting a function — For y = 2x + 1, a table with x = −2, −1, 0, 1, 2 gives y = −3, −1, 1, 3, 5. Plot pairs and join with a straight line because the relation is linear.
y = 2x + 1. Original illustrative model. Full-size diagram

Gradient — Gradient is change in y divided by change in x. From (1, 3) to (4, 9), gradient = (9 − 3)/(4 − 1) = 2. Use a large triangle on a graph for a reliable reading.
Intercept — In y = mx + c, m is gradient and c is the y-intercept because x = 0 gives y = c. For y = −3x + 4, the line falls 3 for each step right and crosses the y-axis at 4.
Rearrange a line — 2x + 3y = 12 becomes 3y = 12 − 2x then y = −(2/3)x + 4. Divide every term by 3, so the gradient is −2/3 and the y-intercept is 4.
Horizontal and vertical — y = 3 is horizontal with zero gradient. x = −2 is vertical and cannot be written as y = mx + c with a finite gradient. Points on it all have x-coordinate −2.
Reading in both directions — To estimate y for a given x, go vertically to the graph then horizontally to the y-axis. To estimate x for a given y, reverse this. A horizontal level may meet a curved graph twice.
Simultaneous equations — The intersection of y = 2x + 1 and y = −x + 7 is approximately (2, 5) on a graph. Its coordinates satisfy both equations. A graph gives estimates limited by its scale and drawing accuracy.

Quadratic graphs — For y = x², x = −2, −1, 0, 1, 2 gives y = 4, 1, 0, 1, 4. Join with a smooth U-shaped curve, not straight segments. The graph is symmetric about the y-axis.
Shifted quadratics — The graph y = x² − 4 has y-intercept −4 and crosses the x-axis at x = −2 and x = 2. At y = 5 it has inputs ±3. State both possible inputs when the context allows them.
y = x² − 4. Original illustrative model. Full-size diagram
Reciprocal context — For a fixed area of 12 m², width w = 12/l. A table gives (l, w) = (1,12), (2,6), (3,4), (4,3). The positive branch curves down and never reaches either axis for positive finite lengths.
Exponential context — For a model population starting at 5 and doubling each hour, N = 5 × 2ᵗ, where t is time in hours and N is population. The power counts the repeated doublings. Values at t = 0, 1, 2, 3 are 5, 10, 20, 40. Read approximate times from a given graph; doubling is not a constant additive rate.

Piecewise graphs — A travel graph may be straight while moving steadily, horizontal while stopped, then straight with a different gradient. On a distance-from-start graph a downward section can represent returning towards the start, not negative elapsed time.
Illustrative journey. Original illustrative model. Full-size diagram
Graph units — On a distance–time graph in km and hours, gradient has units km/h. A steeper line represents greater speed when scales match. A graph with a truncated scale, meaning one that leaves out part of the scale such as starting above zero, can make differences look larger.
Contextual solutions — Where a cost line crosses a budget level gives the maximum affordable amount in the model. If the variable counts whole tickets, a graph reading of 4.8 means at most 4 complete tickets.
Model limitations — A straight-line taxi model may ignore waiting charges; an exponential population model cannot grow forever in a limited habitat. Explain the range over which assumptions are plausible instead of extending a graph without justification.

50 questions · Random sets of 10. These quick checks support revision; practise longer explanations and justified judgements too.