How to graph the function for 0 ≤ x ≤ 2π? If we already know the graph of y = sin x, then with the techniques provided below, the graph of the function can be easily drawn. Let’s first look at the following “transforming rules”.

**[P1] Obtaining the graph of y = sin kx from y = sin x**

The multiplication of the independent variable x by the constant k changes the period of the trigonometric function. While y = sin x has a period of 2π, the trigonometric function y = sin kx has a period of .

Let’s compare the graph of y = sin x (Figure 1a) with y = sin 2x (Figure 1b). The function y = sin x has a period of 2π and y = sin 2x has a period of π.

**Figure 1a**

**Figure 1b**

As another example, the function has period (Figure 1c)

**Figure 1c**

Please take note: If k < 0, apply the formula sin (-α) = – sin α. Thus, to draw the graph of y = sin (-2x), we “recall” that sin (-2x) = – sin 2x, so the graph of y = sin (-2x) will be the same as the graph of y = – sin 2x. (To sketch the graph of y = – sin 2x, please refer to **[P2]** below.) Look at Figure 1d, which is the graph of y = sin (-2x), and compare it with the graph of y = sin 2x shown in Figure 1b.

**Figure 1d**

Similarly, the function y = cos x has a period of 2π while y = cos 2x has a period of π. See Figure 1e below, i.e. the graph of y = cos 2x.

**Figure 1e**

Compare with Figure 1f, which is the graph of the function y = cos x.

**Figure 1f**

**[P2] Obtaining the graph of y = A sin x from y = sin x**

Multiplying sin x by A affects the “height” of the graph. Compare the graph y = sin x with that of y = 2 sin x and y = ½ sin x in Figure 2a.

**Figure 2a**

Please take note: If A < 0, the graph of y = A sin x is the mirror image resulted from the reflection of the graph of y = (-A) sin x about the x-axis. Thus, the graph of y = -2 sin x is the mirror image resulted from the reflection of the graph of y = 2 sin x about the x-axis. See Figure 2b below.

**Figure 2b**

**[P3] Obtaining the graph of y = sin (x±θ) from y = sin x**

The graph of can be obtained by translating the curve as far as π/3 in the direction of the positive x-axis. See Figure 3a.

**Figure 3a**

The graph of can be obtained by translating the curve as far as π/3 in the direction of the negative x-axis. See Figure 3b.

**Figure 3b**

Now, **how to draw y = sin (2x – π/3)?**

The equation y = sin (2x – π/3) can be expressed as y = sin 2(x – π/6). Thus, the curve y = sin (2x – π/3) can be obtained by translating the curve y = sin 2x as far as π/6 in the direction of the positive x-axis. See Figure 3c below.

**Figure 3c**

Similar techniques apply to y = cos (x±θ). Compare the graph of y = cos x with that of y = cos (x-π/4) in Figure 3d.

**Figure 3d**

**[P4] Obtaining the graph of y = b + sin x from y = sin x**

The graph of y = b + sin x is obtained by “raising” or “lowering” the graph of y = sin x as far as b units, parallel to the y-axis. If b > 0 the graph of y = b + sin x is obtained by raising the graph of y = sin x as far as b units in the direction of the positive y-axis. If b < 0 the graph of y = b + sin x is obtained by lowering the graph of y = sin x as far as b units in the direction of the negative y-axis. In Figure 4, it can be seen the effect of adding a constant b to sin x as described above. Notice the difference between the graphs of y = sin x, y = ¾ + sin x, and y = -¾ + sin x.

**Figure 4**

Going back to the question at the beginning of this post, how to graph the function ? First, we need to “trace” the “history” of how the graph of y = sin x changes to that of . It can be illustrated in the following chart.

**Figure 5**

We begin by sketching the graph of y = sin x for 0 ≤ x ≤ 2π. (See Figure 1a.)

**Apply [P1]: The period changes to 2π/3** (See the red curve in Figure 6.)

**Figure 6**

**Apply [P2]: The ‘height” of the curve becomes twice that of the initial height** (See the red curve in Figure 7.)

**Figure 7**

**Apply [P3]:**

y = 2 sin (3x – π/4) can be expressed as y = 2 sin 3(x – π/12). Thus, the curve y = 2 sin (3x – π/4) can be obtained by translating the curve y = 2 sin 3x as far as π/12 in the direction of the positive x-axis. (See the red curve in Figure 8.)

**Figure 8**

**Apply [P4]: Lower the curve as far as 1 unit** (See the red curve in Figure 9.)

**Figure 9**

The graph of y = -1 + 2 sin (3x – π/4) is the red curve in Figure 9. It is resketched in Figure 10.

**Figure 10**