Mathematics (Core)Exam code: 0580
Further Graphs
Coordinate Geometry and Graphs
Substituting carefully
- A table question gives the equation and a row of x values, and asks for the matching y values
- Some entries are already filled in, and those are worth checking against your own working as a test of your method
- Bracket every negative x value as you substitute, then apply the order of operations
- For y = 5 − x² at x = −3: y = 5 − (−3)² = 5 − 9 = −4
- A squared term turns a negative input positive
- On Paper 3, the calculator's table function will generate the whole row at once from the function, the start value, the end value and the step; on Paper 1 every value is worked out by hand
- A reciprocal term has no value at x = 0, so that cell stays empty rather than being filled with a zero
Exam tip
Plot every value from your own table
A table of values with its graph is set in 10 of the 22 papers. All but one of the graph's 3 or 4 marks are for points plotted correctly from your own table, so a wrong value plotted accurately still scores. Plot every pair you have, even one you doubt. Only the correct smooth curve earns the last mark.
Worked example
Filling in a row of values
Fill in the missing y values in the table for y = x² − 3x + 1.
| x | −1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|
| y | 5 | −1 |
Solution:
- At x = 0: 0 − 0 + 1 = 1
- At x = 1: 1 − 3 + 1 = −1
- At x = 3: 9 − 9 + 1 = 1
- At x = 4: 16 − 12 + 1 = 5
- Check a given entry the same way, bracketing the negative: at x = −1, (−1)² − 3 × (−1) + 1 = 1 + 3 + 1 = 5, which matches the table
- The completed row is 5, 1, −1, −1, 1, 5
- The values repeat symmetrically, so the line of symmetry lies half way between x = 1 and x = 2, at x = 1.5
Plotting and drawing
- Plot each pair as a cross, accurate to within half a small square
- Join the points with a single smooth freehand curve, never with a ruler and never as a series of straight segments
- Let the expected shape guide you: a quadratic must be symmetrical, and a reciprocal comes in two separate branches
- A point that sits off the pattern is almost always an arithmetic slip, so recheck it before drawing through it
- Read the axis scales before plotting, since the two axes rarely use the same one