3. Graphs Example 4 Draw a graph of y = x4 –5x3 + 5x2 + 5x – 6 on the domain x = –2
to +3 and use the graph to find the factors and roots of the function. Note: When the function = 0, the roots are where the graph cuts the x-axis. The factors will be (x – roots).
x x4
–5x3 +5x2 +5x –6
y
–2 16 40 20
–10 –6 60
–1 1 5 5
–5 –6 0
y
0 0 0 0 0
–6 –6
1 1
–5 +5 +5 –6 0
2 16
–40 20 10 –6 0
3 81
–135 45 15 –6 0
x
NOTE
From the data above, we can plot the general shape of the graph and find all the roots! If we wished to have a more accurate graph, we would have to find the values of the function (y) for x = –½, ½, 1½, 2½, 3½ and 4½.
The roots are x = –1, 1, 2 and 3. The factors are (x + 1), (x – 1), (x – 2) and (x – 3) Note: If the function does not cut the x-axis at a whole number, the roots (and hence the factors) can only be estimated.
