f (x) = g(x) −x 3 + 4x 2 + x − 4 = x 2 − 2x − 3 x 3 − 3x 2 − 3x + 1 = 0 Graphically: x = −1, 0⋅3, 3⋅7
y
4 5 6 7 8
2 3
1 0.3 –1
–1 –2
–3 –4
1 2 3 3.7 4 x f (x)
g (x)
EXERCISE 8
1. Draw the following cubic functions on graph paper in the given domain for x ∈ R. In each case, write down the roots correct to one decimal place and the point at which the curve crosses the y-axis:
Note: This is also Activity 7. The activity supplies you with the appropriate grids.
(a) f (x) = x 3 − 4x − 1, −2·5 ≤ x ≤ 2·5 (b) f (x) = −x 3 − 5x 2 − x + 8, −5 ≤ x ≤ 1·5 (c) f (x) = x 3 − 3x − 2, −2 ≤ x ≤ 2·5 (d) f (x) = −x 3 − x 2 + 5x − 3, −3 ≤ x ≤ 2 (e) f (x) = x 3 − 2x 2 − x − 2, −1·5 ≤ x ≤ 3
2. Plot the following cubic functions in the given domain for x ∈ R and state how many real roots exist:
(a) y = −x 3 + 2x 2 − x + 2, −1 ≤ x ≤ 3 (b) y = (x + 3)2 (x − 2), −4 ≤ x ≤ 2 (c) y = x 3 − 2x 2 − 5x + 6, −3 ≤ x ≤ 4 (d) y = x 3 − 2x 2 + 3, −1 ≤ x ≤ 3
3. Find the co-ordinates of the points at which the following cubic functions cross the axes:
(a) y = (x + 2)(x − 1)(x − 2) (b) y = −3x(2x − 1)(x + 2) (c) y = x(x − 1)2
