- Coordinates are written (x, y): along the corridor first, then up the stairs.
- The midpoint of two points is the mean of the x values and the mean of the y values.
- The origin is (0, 0); the x-axis is the line y = 0 and the y-axis is x = 0.
- The distance between two points comes from Pythagoras on the horizontal and vertical gaps.
Watch out: Writing the y-coordinate before the x-coordinate, instead of always reading or plotting the x-coordinate first.
- Every straight line can be written y = mx + c, where m is the gradient and c is the y-intercept.
- c is where the line crosses the y-axis; m tells you how steep it is.
- To plot one, make a short table of x and y values, or use the gradient and intercept.
- Rearrange into y = mx + c form before reading off the gradient.
Watch out: Mixing up the gradient and y-intercept when reading y = mx + c, for example giving the y-intercept as the gradient.
- Gradient is change in y divided by change in x (rise over run), read left to right.
- A downhill line has a negative gradient.
- Parallel lines have the same gradient.
- Perpendicular lines have gradients that multiply to -1 (negative reciprocals).
Watch out: Subtracting the coordinates in the wrong order when finding the gradient, which flips the sign of the answer.
- A quadratic graph is a parabola: a U shape when the x squared term is positive, an n shape when negative.
- The roots are where the curve crosses the x-axis, found by solving y = 0.
- The turning point is the vertex; completing the square gives its coordinates.
- The curve is symmetrical about a vertical line through the turning point.
Watch out: Squaring a negative x value incorrectly, for example treating (-2)² as -4 instead of the correct positive value, 4.
- A cubic has an x cubed term and can have up to two turning points and up to three roots.
- A positive cubic runs from bottom-left to top-right; a negative one is the mirror image.
- The reciprocal graph y = 1/x has two separate branches and never touches the axes.
- For a reciprocal graph the axes are asymptotes: the curve gets close but never reaches them.
Watch out: Making an arithmetic error when cubing a negative number, for example treating (-2)³ as 8 instead of the correct value, -8.
- On a distance-time graph the gradient is the speed, and a horizontal line means stopped.
- On a speed-time graph the gradient is the acceleration and the area is the distance travelled.
- A steeper line means a faster rate of change.
- Read the axis labels and units before you interpret anything.
Watch out: Reading a conversion graph in the wrong direction, for example starting from the wrong axis when converting a value.
- The solution to two graphs is where they cross: read off the x and y coordinates.
- To solve f(x) = k, draw the line y = k and find where it meets the curve.
- A curve and a line can cross twice, once, or not at all.
- Rearrange the equation so one side matches a graph you have already drawn.
Watch out: Giving only one intersection point for a quadratic equation, when a curve and a line usually cross at two points, giving two solutions.
- y = f(x) + a moves the graph up by a; y = f(x) - a moves it down.
- y = f(x + a) moves the graph left by a; y = f(x - a) moves it right (inside the bracket does the opposite of what you expect).
- y = -f(x) reflects the graph in the x-axis; y = f(-x) reflects it in the y-axis.
- y = a f(x) stretches the graph vertically by scale factor a.
Watch out: Getting the direction of a horizontal translation backwards, for example thinking y = f(x - 2) shifts the graph to the left instead of to the right.
- The area under a speed-time graph is the distance travelled.
- Split the area into triangles, rectangles and trapeziums, then add them up.
- The trapezium rule estimates the area under a curve using vertical strips.
- Straight-line sections give an exact area; curved sections give an estimate.
Watch out: Forgetting to multiply by h ÷ 2 (half the strip width) when applying the trapezium rule.
