2024 VCE Maths Methods Mini Test 2

Question 1 [2024 Exam 2 Section B Q1]

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

c. Let \( g \) be the function \( g: \mathbb{R} \rightarrow \mathbb{R},\ g(x) = (x + 1)^2(x - 2)^2 \), which is the function \( f \) where \( a = 1 \).

i. Find \( g'(x) \). 1 mark

ii. Find the coordinates of the local maximum of \( g \). 1 mark

iii. Find the values of \( x \) for which \( g'(x) > 0 \). 1 mark

iv. Consider the two tangent lines to the graph of \( y = g(x) \) at the points where \( x =\frac{-\sqrt{3} + 1}{2} \) and \( x = \frac{\sqrt{3} + 1}{2} \). Determine the coordinates of the point of intersection of these two tangent lines. 2 marks

d. Let \( g \) remain as the function \( g: \mathbb{R} \rightarrow \mathbb{R},\ g(x) = (x + 1)^2(x - 2)^2 \), which is the function \( f \) where \( a = 1 \).

Let \( h \) be the function \( h: \mathbb{R} \rightarrow \mathbb{R},h(x) = (x + 1)(x - 1)(x + 2)(x - 2) \), which is the function \( f \) where \( a = -1 \).

i. Using translations only, describe a sequence of transformations of \( h \), for which its image would have a local maximum at the same coordinates as that of \( g \). 1 mark

ii. Using a dilation and translations, describe a different sequence of transformations of \( h \), for which its image would have both local minimums at the same coordinates as that of \( g \). 2 marks