The function \( f \) is defined by \( f(x)=\frac{8}{x^{2}} \). a) Sketch the graph of \( y=f(x) \). b) On a separate set of axes, sketch the graph of \( y=f(x-2) \). Indicate the vertical asymptote and the point where the curve crosses the \( y \)-axis. c) Sketch the graphs of \( y=\frac{8}{x} \) and \( y=\frac{8}{(x-2)^{2}} \) on the same set of axes. Hence state the number of roots of the equation \( \frac{8}{(x-2)^{2}}=\frac{8}{x} \).

To sketch the graph of \( y = f(x) = \frac{8}{x^2} \), we note that it is a rational function where the value of \( f(x) \) approaches infinity as \( x \) approaches zero from either side, creating a vertical asymptote at \( x=0 \). The graph is positive for all \( x \neq 0 \) and has a horizontal asymptote at \( y=0 \) as \( x \) approaches infinity or negative infinity. The curve is symmetrical about the y-axis, indicating that \( f(x) \) is even. When considering \( y = f(x-2) = \frac{8}{(x-2)^2} \), the graph of \( f \) is horizontally shifted 2 units to the right. The vertical asymptote is now at \( x=2 \), and the curve crosses the y-axis when \( x=0 \), giving us \( y = f(-2) = \frac{8}{4} = 2 \). Therefore, the point where the curve intersects the y-axis is at \( (0, 2) \). To sketch \( y = \frac{8}{x} \) and \( y = \frac{8}{(x-2)^2} \) on the same axes, we observe that \( y = \frac{8}{x} \) has a vertical asymptote at \( x=0 \) and a horizontal asymptote at \( y=0 \), while the second function \( y = \frac{8}{(x-2)^2} \) has its vertical asymptote at \( x=2 \) and also approaches zero as \( x \to \pm \infty \). To find the number of roots of the equation \( \frac{8}{(x-2)^{2}} = \frac{8}{x} \), we can simplify this to \( \frac{1}{(x-2)^2} = \frac{1}{x} \) or \( x = (x-2)^2 \). Rearranging the equation gives \( x^2 - 5x + 4 = 0 \), which factors to \( (x-1)(x-4) = 0 \). Thus, there are two roots: \( x=1 \) and \( x=4 \). Since both of these values are defined in both functions’ domains, there are two roots in total.