Problem 53 - Precalculus Essentials

Question

Use a vertical shift to graph one period of the function. $$y=\sin x+2$$

Step-by-Step Solution

Answer

The graph is a sinusoidal curve starting at (0,2), peaking at \(\pi/2, 3\), crossing back at \(\pi, 2\), bottoming at \(3\pi/2, 1\), and finishing the cycle at \(2\pi, 2\). This pattern repeats every \(2\pi\). The '+2' in the function resulted in this graph being 2 units higher than the basic sine graph.

Step 1: Understand the Basic Sine Function

The basic sine function, \(y = \sin x\), starts at (0,0), rises to a maximum at \(\pi/2\), descends to 0 at \(\pi\), reaches a minimum at \(3\pi/2\), and returns to 0 at \(2\pi\). This cycle repeats every \(2\pi\) units and is called the 'period' of the function. These key points will be the same for \(y = \sin x + 2\), but every y-coordinate will be shifted up by 2 units.

Step 2: Plot Key Points with Vertical Shift

Now plot the key points of the basic sine function, but add 2 to every y-coordinate because of the '+2' in our function. The key points for one period are thus: (0,2), \(\pi/2, 3\), \(\pi, 2\), \(3\pi/2, 1\), and \(2\pi, 2\).

Step 3: Draw the Graph

Connect these points smoothly, resulting in a wave-like shape that rises to a maximum at \(\pi/2\), descends to a minimum at \(3\pi/2\), and returns to the midpoint at \(2\pi\). Make sure the graph looks symmetrical since the sine function is symmetrical.