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

To shift the hyperbola \( f(x) = \frac{3}{x-7} - 4 \) two units to the left, you replace \( x \) with \( x + 2 \). Then, to shift it 3 units up, you add 3 to the entire function. The resulting equation becomes \( g(x) = \frac{3}{(x + 2) - 7} - 4 + 3 \), simplifying to \( g(x) = \frac{3}{x - 5} - 1 \). For the second transformation, shifting 1 unit right involves replacing \( x \) with \( x - 1 \), then for the 2 units down, you subtract 2 from the entire function. This results in \( h(x) = \frac{3}{(x - 1) - 7} - 4 - 2 \), simplifying to \( h(x) = \frac{3}{x - 8} - 6 \). Now, to achieve the new asymptotes at \( x = -4 \) and \( y = -1 \), we can directly adjust the function. The new equation would take the form \( j(x) = \frac{3}{x + 4} - 1 \).