QUESTION 3 The equation of a hyperbola is given by \( f(x)=\frac{3}{x-7}-4 \). Write down the equation of the new function that is formed when \( f \) is transformed as follows 3.1 Shift noo units to the left 3.2 Shif 3 units up 3.3 Shift 1 unit nght and 2 units down \( 3.4 \quad \) The equation of the new hypertola has new asymplotes at \( x-4 \) and \( y=-1 \)

Una sucesión infinita tiene la caracteristica de que: a. El término \( a_{n} \) siempre es el último de la sucesión. b. Cada término \( a_{n} \) tiene un antecesor igual a \( a_{n+1} \) c. Cada término \( a_{n} \) es el último en sucesiones pares. d. Cada término \( a_{n} \) tiene un sucesor igual a \( a_{n+1} \)

(a) Montrer que \( \left(v_{n}\right) \) est une suite géométrique de raison \( \mathrm{q}=4 \) (b) Écrire \( v_{n} \) puis \( u_{n} \) en fonction de \( n \) C) Calculer en fonction de \( n: S_{n}=\sum_{k=1}^{n} v_{k} \) puis \( T_{n}=\prod_{k=1}^{n} v_{k} \) On considère la suite \( \left(u_{n}\right) \) définie par: \( u_{0}=2 e t(\forall n \in \mathbb{N}) ; u_{n+1}=\sqrt{6+u_{n}} \) 1 Montrer que : \( (\forall n \in \mathbb{N}) ; 1<u_{n}<3 \) 2 Vérifier que \( :(\forall n \in \mathbb{N}) ; 3-u_{n+1}=\frac{1}{3+\sqrt{6+u_{n}}}\left(3-u_{n}\right) \)

The population of a certain species in a protected area can be modeled by the function \( P(t) = 200 + 50 \sin(t) \), where \( t \) is measured in years. Determine the total population increase over one complete cycle (from \( t=0 \) to \( t=2\pi \)).

1. Expand \( (1+x)^{\frac{1}{3}} \) in ascending powers of x up to the term \( x^{3} \). By substituting 0.08 for \( x \) in your result. Obtain an approximation value of the cube root of 5 .

QUESTION 4 It is given that the asymptotes of \( f(x)=\frac{6}{x+p}+q \) intersect at \( (4 ; 3) \). \( 4.1 \quad \) Write down the equation of \( f \). \( 4.2 \quad \begin{array}{l}\text { Sketch the graph of } f \text {, clearly showing all the intercepts with the axes and any } \\ \text { asymptotes. }\end{array} \) \( 4.4 \quad \begin{array}{l}g \text { is one of the axes of symmetry of } f \text { and it is a decreasing function. Determine the } \\ \text { equation of } g \text {. } \\ 4.5 \\ (-3 ; 2) \text { is a point on } f \text {. Determine the coordinates of the image of this point after } \\ \text { reflection in } g \text {. }\end{array} \)