Problem 35 - Precalculus: A Concise Course

Question

Graph \(f\) and \(g\) on the same set of coordinate axes. (Include two full periods.) $$ \begin{array}{l} f(x)=-\frac{1}{2} \sin \frac{x}{2} \\ g(x)=3-\frac{1}{2} \sin \frac{x}{2} \end{array} $$

Step-by-Step Solution

Answer

Graphing function \(f(x)\) shows a wave oscillating between -1/2 to 1/2 with a period of \(4π\). Graphing function \(g(x)\) shows a wave similar to \(f(x)\) but shifted up by 3 units, oscillating between 2.5 and 3.5 with the same period of \(4π\).

Step 1: Identifying the parameters of the sin functions

Recognize that both \(f(x)\) and \(g(x)\) are transformations of the basic sin function. For \(f(x)=-\frac{1}{2}sin(\frac{x}{2})\), the amplitude (shown by the coefficient of the sin function) is -1/2, meaning the graph will fluctuate between -1/2 and 1/2. The period (associated with the coefficient of x inside the sin function) is \(2π*2=4π\). For \(g(x)=3-\frac{1}{2}sin\frac{x}{2}\), the amplitude is the same, but the graph is vertically shifted up by 3 units. It fluctuates between \(3-\frac{1}{2} = 2.5\) and \(3+\frac{1}{2} = 3.5\). The period is the same, \(4π\).

Step 2: Graph the functions

Now that the amplitude and period of the functions have been established, all that remains is to draw them on the same set of axes. The axes should be large enough to capture two full periods, meaning they must extend at least to \(4π\). For \(f(x)\), start at the origin, reach the minimum at \(π\), return to 0 at \(2π\), reach the maximum at \(3π\), and return to 0 at \(4π\). Repeat for another period. For \(g(x)\), the graph behaves the same but it starts at 3, reaches a minimum at \(2.5\) (\(π\)), returns to 3 (\(2π\)), reaches a max at \(3.5\) (\(3π\)), returns to 3 (\(4π\)). Repeat for another period. Please remember to label the function lines and their significant points. Lastly, to make the graph clearer, like periodic peaks and zeros.