GCSE Maths · Quadratic and Other Graphs · Foundation · Summary
GCSE Maths Quadratic and Other Graphs (Foundation): summary and key words
Part of the GCSE Maths quadratic and other graphs chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page
Chapter summary
| Skill | Method |
|---|---|
| Table of values | Substitute each x into the rule (negatives in brackets), write the y-values, plot (x, y), join with a smooth curve |
| Quadratic shape | x² positive: U-shape with a minimum. x² negative: n-shape with a maximum. Symmetrical about a vertical line |
| Roots, y-intercept | Roots: where the curve meets the x-axis (y = 0). y-intercept: where it meets the y-axis (x = 0) |
| Turning point and symmetry | Turning point (x, y) is the lowest or highest point. Line of symmetry x = x-value of the turning point, halfway between the roots |
| Roots by algebra | (x + p)(x + q) = 0: each bracket equal to 0 in turn (signs flip). x² = k gives x = ±√k |
| Solving by graph | (rule) = k: draw y = k and read the x-values. (rule 1) = (rule 2): read the x-values where the two graphs cross |
| Cubic graphs | Highest power x³. y = x³ is S-shaped through the origin; negative x³ flips it. A cubic can cross the axis up to 3 times |
| Reciprocal graphs | y = k ÷ x: two branches, no point at x = 0. Positive k: top-right and bottom-left. Negative k: top-left and bottom-right |
| Recognising graphs | Highest power x: straight line. x²: parabola. x³: cubic. x on the bottom: reciprocal. Use the sign and the y-intercept |
| Real-life graphs | Check the axes and units. Read values with straight lines to the curve. Maximum = turning point. A horizontal line can meet the curve twice |
| Distance-time graph | Gradient = speed = change in distance ÷ change in time. Flat = stopped. Average speed = total distance ÷ total time |
| Speed-time graph | Gradient = acceleration (m/s²). Flat = constant speed. Area under the graph = distance travelled |
| Completing the square (H) | x² + bx + c = (x + b/2)² + c − (b/2)². Turning point (−b/2, c − b²/4) |
| Sketching a quadratic (H) | Shape, y-intercept, roots, turning point, with coordinates labelled and a smooth symmetrical curve |
Can you…?
- I can complete a table of values for a quadratic and plot it as a smooth curve.
- I can say whether a quadratic is U-shaped or n-shaped from its equation.
- I can read the roots, y-intercept, line of symmetry and turning point from a quadratic graph.
- I can find roots by algebra when the rule is in brackets or is x² minus a number.
- I can solve an equation by reading a graph, including by drawing a horizontal line or a second graph.
- I can draw and recognise cubic and reciprocal graphs.
- I can match a graph to its equation from its shape.
- I can read values, maximum values and roots from real-life curved graphs.
- I can read distances, times and speeds from a distance-time graph and find an average speed.
- I can find acceleration and distance travelled from a speed-time graph.
- Higher: I can sketch a quadratic and find its turning point by completing the square.
Key terms
- Quadratic
- A rule whose highest power of x is x². Its graph is a parabola.
- Parabola
- The U-shaped or n-shaped curve that is the graph of a quadratic.
- Table of values
- A list of x-values with the y-values worked out from the rule.
- Root
- An x-value where a graph meets the x-axis (where y = 0).
- y-intercept
- The point where a graph crosses the y-axis (where x = 0).
- Turning point
- The lowest point (minimum) or highest point (maximum) of a curve.
- Line of symmetry
- A mirror line that splits a graph into two matching halves. For a quadratic it is vertical through the turning point.
- Cubic
- A rule whose highest power of x is x³.
- Reciprocal
- One divided by a number. The graph of y = k/x has two branches.
- Asymptote
- A line that a curve gets closer to but never touches.
- Sketch
- A quick drawing of a graph that shows its shape and key points without an accurate scale.
- Gradient
- How steep a line is: change in y ÷ change in x.
- Speed
- Distance travelled per unit time. The gradient of a distance-time graph.
- Acceleration
- The rate at which speed changes. The gradient of a speed-time graph, in m/s².
- Deceleration
- Slowing down: a negative acceleration.
- Completing the square
- Writing x² + bx + c as (x + p)² + q. (Higher)
- Surd
- An exact root such as √5 that is not a whole number or fraction. (Higher)