MathematicsExam code: 0580

Further Graphs and Tangents

Coordinate Geometry and Graphs

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 − 2x² at x = −3: y = 5 − 2 × (−3)² = 5 − 2 × 9 = −13
A squared term turns a negative input positive, while a cubed term keeps it negative
The calculator's table function will generate the whole row at once from the function, the start value, the end value and the step
A reciprocal term has no value at x = 0, so that cell stays empty rather than being filled with a zero

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

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

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