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 \( t=2\pi \)).

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: \( \begin{array}{ll}3.1 & \text { Shift two units to the left } \\ 3.2 & \text { Shift } 3 \text { units up } \\ 3.3 & \text { Shift } 1 \text { unit right and } 2 \text { units down } \\ 3.4 & \text { The equation of the new hyperbola has new asymptotes at } x=-4 \text { and } y=-1\end{array} \) (1)

etch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x-1}+1 \) early indicate all intercepts with the axes, turning point(s) and asymptote(s).

If \( f(x)=\frac{\sqrt{x+2}}{3-3 x^{2}} \), for which values of \( x \) is 1.2.2 \( \quad f(x) \) non real.

Graph the function \[ f(x)=\left\{\begin{array}{ll}\lfloor x\rfloor, & x \geq 0 \\ \lceil x\rceil, & x<0\end{array}\right. \]

\( \left. \begin{array} { l } { \log _ { 3 } 2 = a } \\ { \log _ { 3 } 5 = b } \end{array} \right. \left\{ \begin{array} { l } { \log _ { 25 } 16 = ? } \end{array} \right. \)