VCE Methods Functions Application Task 1
Number of marks: 7
Reading time: 1 minute
Writing time: 10 minutes
Section B – Calculator Allowed
Instructions
• Answer all questions in the spaces provided.
• Write your responses in English.
• In questions where a numerical answer is required, an exact value must be given unless otherwise specified.
• In questions where more than one mark is available, appropriate working must be shown.
• Unless otherwise indicated, the diagrams in this book are not drawn to scale.
Consider the function \( f: \mathbb{R} \rightarrow \mathbb{R},\ f(x) = (x + 1)(x + a)(x - 2)(x - 2a) \) where \( a \in \mathbb{R} \).
a. State, in terms of \( a \) where required, the values of \( x \) for which \( f(x) = 0 \). 1 mark
b. Find the values of \( a \) for which the graph of \( y = f(x) \) has
i. exactly three x-intercepts. 2 marks
ii. exactly four x-intercepts. 1 mark
The points shown on the chart below represent monthly online sales in Australia.
The variable \( y \) represents sales in millions of dollars.
The variable \( t \) represents the month when the sales were made, where \( t = 1 \) corresponds to January 2021, \( t = 2 \) corresponds to February 2021 and so on.
a. A cubic polynomial \( p : (0, 12] \rightarrow \mathbb{R}, \, p(t) = at^3 + bt^2 + ct + d \) can be used to model monthly online sales in 2021.
The graph of \( y = p(t) \) is shown as a dashed curve on the set of axes above.
It has a local minimum at (2, 2500) and a local maximum at (11, 4400).
i. Find, correct to two decimal places, the values of \( a \), \( b \), \( c \) and \( d \). 3 marks
End of examination questions
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