Further Graphical TechniquesEdexcel GCSE Maths: Revision notes
Section 1
What Do Quadratic, Cubic and Reciprocal Graphs Look Like?
- A quadratic function y = ax² + bx + c produces a parabola (a symmetric U-shape or ∩-shape). Its roots are the x-values where y = 0, found by solving the quadratic. Its turning point (vertex) can be found by completing the square, and the graph has a line of symmetry passing through the turning point.
- A cubic function such as y = x³ produces an S-shaped curve, which can have up to three roots and up to two turning points.
- The reciprocal function y = 1/x produces two separate curved branches that get closer and closer to the x-axis and y-axis without ever touching them — these lines are called asymptotes.
y = x² − 4 has roots at x = 2 and x = −2, and a turning point (minimum) at (0, −4).
Section 2
Exponential and Trigonometric Graphs
- The exponential function y = kˣ (for positive k) always passes through (0, 1) since k⁰ = 1. It increases rapidly as x increases (for k > 1), and approaches but never reaches y = 0 as x decreases — the x-axis is an asymptote.
- The graphs of y = sin x and y = cos x oscillate smoothly between −1 and 1, repeating every 360°.
- The graph of y = tan x repeats every 180° and is undefined at 90°, 270°, and so on, where the function shoots off towards positive or negative infinity.
Remember the period of sin x and cos x is 360°, but the period of tan x is only 180°.
Section 3
Translating and Reflecting a Graph
Starting from a known function f(x), you can describe new graphs algebraically without needing to draw anything:
| Transformation | Effect |
|---|---|
| f(x + a) | shifts the graph left by a (note the sign is opposite to what you might expect) |
| f(x) + a | shifts the graph up by a |
| f(−x) | reflects the graph in the y-axis |
| −f(x) | reflects the graph in the x-axis |
Example: if f(x) = x², then f(x) + 3 = x² + 3 is the same curve shifted up by 3, and −f(x) = −x² is the same curve reflected in the x-axis (turned upside down).
f(x + 2) shifts a graph LEFT by 2, not right — students very often get this sign the wrong way round.
Section 4
Stretching a Graph
Multiplying inside or outside the function produces a stretch rather than a shift:
- f(ax) stretches the graph horizontally by scale factor 1/a (parallel to the x-axis)
- af(x) stretches the graph vertically by scale factor a (parallel to the y-axis)
Example: if f(x) = x², then f(2x) = (2x)² = 4x² is a horizontal stretch by scale factor 1/2, while 2f(x) = 2x² is a vertical stretch by scale factor 2.
Section 5
Estimating the Gradient of a Curve
A curve doesn't have one constant gradient like a straight line does — its steepness changes from point to point. To estimate the gradient at a specific point on a curve:
- Draw a tangent to the curve at that point (a straight line that just touches the curve there without crossing it)
- Choose two clear points on the tangent line
- Calculate the gradient of the tangent using (change in y) ÷ (change in x)
This gradient is only an estimate, because it depends on how accurately the tangent is drawn.
In context questions (e.g. a distance-time graph), the gradient of a tangent to a curve gives the instantaneous rate of change at that instant, such as speed at a specific moment.
Must Know
- Quadratics graph as parabolas with roots, a turning point and a line of symmetry; cubics are S-shaped; y = 1/x has two branches and asymptotes at both axes
- y = kˣ passes through (0, 1) and never reaches y = 0
- sin x and cos x repeat every 360°; tan x repeats every 180° and is undefined every 90° + 180°n
- f(x + a) shifts left by a; f(x) + a shifts up by a; f(−x) reflects in the y-axis; −f(x) reflects in the x-axis
- f(ax) stretches horizontally by scale factor 1/a; af(x) stretches vertically by scale factor a
- To estimate a curve's gradient at a point, draw a tangent there and find its gradient
That's the notes covered.
Carry on to the next subtopic.