Problem 53 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The graph of the given function, y = sin(x) + 2, is a sine wave that peaks at 3 and troughs at 1, crossing the x-axis at \( \pi/2, \pi, 3\pi/2 \) and \(2\pi\). The function repeats its pattern every \(2\pi\) units observable as the periodic nature.

Step 1: Draw the Original Function

First, make a rough sketch of the original function y = sin(x) over its period, which is \(0\) to \(2\pi\). This graph oscillates between 1 and -1, crossing the x-axis at \( \pi/2, \pi, 3\pi/2 \) and \(2\pi\).

Step 2: Apply the Vertical Shift

On this same sketch, place dots 2 units above each significant point (maxima, minima, x-intercepts). Then, join these points to form a sine wave, this is the graph of y = sin(x) + 2. The graph still crosses the x-axis at the same points, but the maximum is now 3 and minimum is now 1.

Step 3: Check the Period

Lastly, ensure that the period still holds, which means that the pattern repeats every \(2\pi\) units. Notice how each up and down wave (or cycle) of the sinusoidal graph is identical. This repetition every \(2\pi\) units is precisely what defines the function as periodic.