0580

Further Graphs and Tangents

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 − 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
Exam tip

"Complete the table" appears in 27 of the 43 papers, and the entries already filled in are there to be used: work one of them out yourself first. If your value matches theirs your method is right, and if it does not you have found the mistake before it reaches the graph and everything that follows it.

Worked example

Filling in a row of values

Complete the table of values for y = x² − 3x + 1.

x−101234
y1−1

Solution:

  • At x = −1: (−1)² − 3 × (−1) + 1 = 1 + 3 + 1 = 5
  • At x = 1: 1 − 3 + 1 = −1
  • At x = 3: 9 − 9 + 1 = 1
  • At x = 4: 16 − 12 + 1 = 5
  • The completed row is 5, 1, −1, −1, 1, 5
  • The given entries at x = 0 and x = 2 match, which confirms the method
  • 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