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: \( 3.1 \quad \) Shift two units to the left \( 3.2 \quad \) Shift 3 units up \( 3.3 \quad \) Shift 1 unit right and 2 units down \( 3.4 \quad \) The equation of the new hyperbola has new asymptotes at \( x=-4 \) and \( y=-1 \)

Sure! Let’s break down the transformations and find the new equations step by step. **3.1** Shifting the function \( f \) two units to the left means replacing \( x \) with \( x + 2 \): \[ f(x) = \frac{3}{(x + 2) - 7} - 4 = \frac{3}{x - 5} - 4 \] **3.2** Shifting the function 3 units up involves adding 3 to the entire function: \[ f(x) = \frac{3}{x - 5} - 4 + 3 = \frac{3}{x - 5} - 1 \] **3.3** Shifting the function 1 unit right and 2 units down translates to replacing \( x \) with \( x - 1 \) and subtracting 2 from the function: \[ f(x) = \frac{3}{(x - 1) - 5} - 1 - 2 = \frac{3}{x - 6} - 3 \] **3.4** For the new hyperbola with asymptotes at \( x = -4 \) and \( y = -1 \), we can represent it in the standard form of a hyperbola that has vertical and horizontal asymptotes: The general form will be \( \frac{(y + 1)^2}{a^2} - \frac{(x + 4)^2}{b^2} = 1 \). You can determine \( a \) and \( b \) based on specific characteristics. The equation could look something like: \[ \frac{(y + 1)^2}{1} - \frac{(x + 4)^2}{1} = 1 \] which can be simplified based on the specific context. And voilà! You’ve got an engaging journey through transformations of a hyperbola.