The shapes of the standard graphs, and the key points a sketch must show: where it crosses each axis and where it turns.
What you will learn
- Recognise a straight line, a quadratic, a cubic and a reciprocal graph from its equation
- Find where a graph crosses the x-axis and the y-axis
- Find the turning point of a quadratic by symmetry or by completing the square
- Choose or draw a sketch that shows the key points
Know the shapes
A sketch is not plotted point by point, but its shape must be right.
Look at the highest power of x to tell the type:
- y = mx + c (highest power 1): a straight line, with the same gradient everywhere.
- y = x² + ... (a quadratic): a U shape (∪) with one turning point, its lowest point (a minimum). With −x² it is upside down (∩) and the turning point is the highest point (a maximum). A quadratic has a vertical line of symmetry through its turning point; for y = x² + c that line is the y-axis.
- y = x³ + ... (a cubic): an S shape, from bottom left to top right. With −x³ it falls from top left to bottom right. A cubic has no line of symmetry (y = x³ has rotational symmetry about the origin).
- y = k/x (a reciprocal graph): two separate curves that get close to the axes but never touch them, so it never crosses either axis.
Two brackets multiplied, such as (x + 1)(x − 5), make an x² term, so the graph is a quadratic. Three brackets make a cubic.
A straight line ax + by = c is quickest to sketch from its two intercepts (the points where it crosses the axes). On the x-axis y = 0, and on the y-axis x = 0. For 2x + 5y = 10: y = 0 gives x = 5, so (5, 0); x = 0 gives y = 2, so (0, 2). Join them with a ruler and extend the line.
The rest of this lesson and its 11 practice questions are locked.
In this lesson
- Know the shapes
- The key points
- Turning points by completing the square
- Try it: Sketch and check
11 practice questions, marked as you go; about 25 minutes in all.
Key terms
Turning point, Minimum, Maximum, Intercepts
In this unit: Drawing and constructing
The answers papers ask you to draw: graphs plotted from a table, sketches with their key points, statistical charts, ruler-and-compass constructions and loci, each checked the way a mark scheme checks it.