CUBIC FUNCTIONS
A third degree polynomial is called a cubic and is a function f , with rule
f(𝑥) = 𝑎𝑥 3 + 𝑏𝑥 2 + 𝑐𝑥 + 𝑑, 𝑎 ≠ 0.
1. The following diagram shows the graph of 𝑦 = 𝑥 3 + 2𝑥 2 − 5𝑥 − 6.
(i)
(ii)
Use the graph to solve the equations
(a) 𝑥 3 + 2𝑥 2 − 5𝑥 − 6 = 0
[2]
(b) 𝑥 3 + 2𝑥 2 − 5𝑥 − 6 = 2𝑥 + 2
[3]
Calculate an estimate of the
(a) Gradient of the curve at the point where 𝑥 = 2
[2]
(b) Area bounded by the curve, 𝑥 = 0, 𝑦 = 0 and 𝑦 = 20
[2]
2. The following diagram shows the graph of 𝑦 = 𝑥 3 + 𝑥 2 − 5𝑥 + 3.
1|Page
(i)
Use the graph to solve the equations
(a) 𝑥 3 + 𝑥 2 − 5𝑥 + 3 = 0
[2]
(b) 𝑥 3 + 𝑥 2 − 5𝑥 + 3 = 𝑥 + 3
[3]
(ii)
Calculate an estimate of the
(a) Gradient of the curve at the point where 𝑥 = −2
[2]
(b) Area bounded by the curve, 𝑥 = −2, 𝑥 = 0 and 𝑦 = −10
[2]
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3. The following diagram shows the graph of 𝑦 = 𝑥 2 (5 - 𝑥).
(i)
(ii)
3|Page
Use the graph to solve the equations
(a) 𝑥 2 (5 − 𝑥). = 0
[2]
(b) 𝑥 2 (5 − 𝑥). = −𝑥 + 5
[3]
Calculate an estimate of the
(a) Gradient of the curve at the point where 𝑥 = 4
[2]
(b) Area bounded by the curve, 𝑥 = 1, 𝑥 = 4 and 𝑦 = 0
[2]