QUESION 6 NOR Given: \( f(x)=\frac{2}{x-1}-2 \) 4.1 Write down the equations of the asymptote 4.3 Draw a graph of \( f \). Clearly label ALL the asymptotes on your graph. Fotermine the equation of the line of symm 4.4

To find the equations of the asymptotes of the function \( f(x) = \frac{2}{x-1} - 2 \), we need to consider the vertical asymptote and the horizontal asymptote. 1. **Vertical Asymptote:** The vertical asymptote occurs when the denominator of the function is equal to zero. In this case, the denominator is \( x - 1 \). Therefore, the vertical asymptote is at \( x = 1 \). 2. **Horizontal Asymptote:** To find the horizontal asymptote, we need to consider the behavior of the function as \( x \) approaches infinity. Since the function is a rational function, the horizontal asymptote is determined by the degree of the numerator and the denominator. In this case, the degree of the numerator is 0 and the degree of the denominator is 1. Therefore, the horizontal asymptote is the line \( y = 0 \). Now, let's draw the graph of \( f(x) \) and label the asymptotes. The graph of \( f(x) \) will have a vertical asymptote at \( x = 1 \) and a horizontal asymptote at \( y = 0 \). To find the equation of the line of symmetry, we need to find the axis of symmetry of the graph. The axis of symmetry is the vertical line that passes through the vertex of the graph. Since the graph is a rational function, the axis of symmetry is the vertical line \( x = 1 \). Therefore, the equation of the line of symmetry is \( x = 1 \).