Sketching Cubic Graphs (Grade 12) — Worksheet & Memo

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Grade 12 · Calculus

Sketching Cubic Graphs (Grade 12)

MEMORANDUM

Name:

Class:

Date:

Mark

/ 16

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Determine the y-intercept of f(x) = x³ − 3x² − 9x + 5.
(2)
f(0) = 5✓✓ (2)
2
Determine the coordinates of the stationary points of f(x) = x³ − 3x² − 9x + 5.
(2)
f′(x) = 3(x−3)(x+1) = 0 → (−1 ; 10) and (3 ; −22)✓✓ (2)
3
State, with a reason, which stationary point is the local maximum.
(2)
(−1 ; 10), because f″(−1) = −12 < 0 (concave down).✓✓ (2)
4
Determine the coordinates of the point of inflection.
(2)
f″(x) = 6x − 6 = 0 → x = 1 → (1 ; −6)✓✓ (2)
5
For which values of x is f decreasing?
(2)
−1 < x < 3✓✓ (2)
6
For which values of x is f concave up?
(2)
f″(x) > 0 → x > 1✓✓ (2)
7
Where does the point of inflection of a cubic always lie relative to the turning points?
(2)
Exactly midway between them.✓✓ (2)
8
Determine the stationary points of g(x) = x³ − 12x.
(2)
g′ = 3x² − 12 = 0 → x = ±2 → (2 ; −16) and (−2 ; 16)✓✓ (2)

TOTAL: 16 marks