Sketching Cubic Graphs (Grade 12) — Worksheet & Memo

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Grade 12 · Calculus

Sketching Cubic Graphs (Grade 12)

EXAM-STYLE CLASS TEST

Marks

Duration

50 minutes

Questions

Name:

Class:

Date:

Mark

/ 32

Instructions and Information

Answer ALL the questions in this question paper.
Answer QUESTION 1 by circling the letter (A–D) in the answer grid at the end of that section.
Show ALL calculations clearly.
Show all units where applicable.
Number the answers correctly according to the numbering system used in this question paper.
A non-programmable calculator may be used, unless stated otherwise.
Write neatly and legibly.

Question 1

[10 MARKS]

Four options are given as possible answers to the following questions. Choose the correct answer and circle the letter (A–D) in the grid at the end of this section. If you want to change your choice, put a cross through the wrong letter and circle your new choice.

1.1
At a STATIONARY point of a cubic …
(1)
A)f'(x) = 0
B)f(x) = 0
C)f''(x) = 0
D)x = 0
1.2
h(x) = ax³ + bx² has turning points at (0 ; 0) and (4 ; 32). Then …
(1)
A)a = −1 and b = 6
B)a = 1 and b = −6
C)a = −1 and b = −6
D)a = 6 and b = −1
1.3
For h(x) = −x³ + 6x², the x-intercept other than the origin is at …
(1)
A)x = 6
B)x = 4
C)x = −6
D)x = 32
1.4
For h(x) = −x³ + 6x², h is INCREASING for …
(1)
A)0 < x < 4
B)x < 0 or x > 4
C)x > 4
D)x < 0
1.5
For h(x) = −x³ + 6x², h is CONCAVE DOWN for …
(1)
A)x > 2
B)x < 2
C)x > 4
D)0 < x < 4
1.6
The POINT OF INFLECTION of a cubic is where …
(1)
A)f''(x) = 0
B)f'(x) = 0
C)f(x) = 0
D)the graph cuts the y-axis
1.7
A cubic whose coefficient of x³ is POSITIVE …
(1)
A)rises to the right and falls to the left
B)falls to the right and rises to the left
C)is increasing everywhere
D)has no turning points
1.8
How many turning points can a cubic graph have?
(1)
A)at most two
B)exactly two
C)exactly one
D)at most three
1.9
For f(x) = x³ − 3x², the LOCAL MINIMUM is at …
(1)
A)(2 ; −4)
B)(0 ; 0)
C)(2 ; 4)
D)(−2 ; −20)
1.10
To decide whether a stationary point is a maximum or a minimum, you can …
(1)
A)test the sign of f''(x) at that point
B)test the sign of f(x) at that point
C)check whether x is positive
D)count the x-intercepts

Answer grid — circle your answers for Question 1

1.1	A	B	C	D
1.2	A	B	C	D
1.3	A	B	C	D
1.4	A	B	C	D
1.5	A	B	C	D
1.6	A	B	C	D
1.7	A	B	C	D
1.8	A	B	C	D
1.9	A	B	C	D
1.10	A	B	C	D

Question 2

[12 MARKS]

Given f(x) = −x³ + 6x².

2.1
Calculate the coordinates of the x-intercepts of f.
(4)
2.2
Determine the coordinates of the stationary points of f.
(5)
2.3
Determine the coordinates of the point of inflection.
(3)

Question 3

[10 MARKS]

Answer the questions below about the same graph.

3.1
Write down the values of x for which f is concave up.
(2)
3.2
Determine the equation of the tangent to f at x = 1.
(4)
3.3
For which values of k will −x³ + 6x² = k have THREE distinct real roots? Justify your answer.
(4)

TOTAL: 32 marks

This question paper consists of 3 questions.