Question Three \( y=\frac{2}{4} x^{3}-3 x^{2}-6 x+9 \) - For the function \( y-2=2^{-x} \), sketch its graph clearly showing the intercepts ar asymptote \( [5 \mathrm{mks}] \)

To sketch the graph of the function \( y - 2 = 2^{-x} \), first isolate \( y \) to get \( y = 2^{-x} + 2 \). The intercept occurs when \( x=0 \), giving \( y = 3 \), so the y-intercept is at point (0, 3). The asymptote here is horizontal, positioned at \( y = 2 \), where the graph approaches but never touches as \( x \) increases. You can easily plot the key points by calculating few values of x, such as \( x = -1 \) giving \( y = 3 \) and \( x = 1 \) resulting in \( y = 2.5 \). To further enhance your graph sketching skills, remember to analyze the behavior of the function in the long term. As \( x \) approaches infinity, \( 2^{-x} \) approaches zero, thus reinforcing the horizontal asymptote at \( y = 2 \). Conversely, as \( x \) heads toward negative infinity, \( y \) increases significantly, allowing you to illustrate the function's increasing nature while it trends down toward its horizontal asymptote. Having this understanding will help you portray a much clearer and more accurate graph!