Graphs of Trigonometric Functions

Trigonometric functions are periodic, meaning their graphs repeat in regular intervals or cycles. Understanding the basic shapes of sine, cosine, and tangent graphs, as well as how to transform them, is an essential skill in trigonometry.

The Basic Trigonometric Graphs

Before applying transformations, you must know the "parent" graphs of the three primary trigonometric functions:

• Sine ($y = \sin x$): Starts at the origin $(0,0)$, rises to a maximum of $1$ at $\frac{\pi}{2}$, crosses the x-axis at $\pi$, falls to a minimum of $-1$ at $\frac{3\pi}{2}$, and completes its cycle at $2\pi$. Its period is $2\pi$.
• Cosine ($y = \cos x$): Starts at a maximum of $(0,1)$, crosses the x-axis at $\frac{\pi}{2}$, falls to a minimum of $-1$ at $\pi$, crosses the x-axis again at $\frac{3\pi}{2}$, and completes its cycle at $2\pi$. Its period is $2\pi$.
• Tangent ($y = \tan x$): Passes through the origin but has vertical asymptotes at $x = \frac{\pi}{2} + n\pi$ (where $n$ is an integer) because tangent is undefined at these points. Its period is $\pi$.

Transformations: The General Form

We can transform these basic graphs using the general equation:

$y = A\sin(Bx - C) + D$

(This same format applies to cosine and tangent functions).

Each constant alters the graph in a specific way:

• Amplitude ($|A|$): The height from the center line to the peak (or trough). If $A$ is negative, the graph reflects across the midline. (Note: Tangent functions do not have an amplitude since they extend infinitely up and down).
• Period: The length of one complete cycle.
  • For sine and cosine: $\text{Period} = \frac{2\pi}{|B|}$
  • For tangent: $\text{Period} = \frac{\pi}{|B|}$
• Phase Shift ($\frac{C}{B}$): The horizontal shift of the graph. If $\frac{C}{B}$ is positive, the graph shifts to the right. If negative, it shifts to the left.
• Midline ($y = D$): The vertical shift. This is the new horizontal center line of the graph.

Example Problems

Example 1: Graphing a Transformed Sine Function

Problem: Graph $y = 3\sin(2x - \frac{\pi}{3}) + 1$ over two periods.

Solution: Let's identify the transformation variables from the equation $A=3$, $B=2$, $C=\frac{\pi}{3}$, and $D=1$.

1. Amplitude: $|A| = 3$. The graph will reach $3$ units above and below the midline.
2. Midline: $y = 1$. The maximum value is $1 + 3 = 4$, and the minimum is $1 - 3 = -2$.
3. Period: $\frac{2\pi}{B} = \frac{2\pi}{2} = \pi$. One full cycle takes $\pi$ units.
4. Phase Shift: $\frac{C}{B} = \frac{\pi/3}{2} = \frac{\pi}{6}$. The graph starts its standard sine cycle at $x = \frac{\pi}{6}$ instead of $x = 0$.

To graph, start at the phase shift $x = \frac{\pi}{6}$ on the midline $y = 1$. Follow the sine pattern (midline, max, midline, min, midline) ending the first cycle at $x = \frac{\pi}{6} + \pi = \frac{7\pi}{6}$. Repeat this pattern for a second period.

Example 2: Finding the Equation from Properties

Problem: Find the equation of a cosine function with amplitude $4$, period $\pi$, and shifted right $\frac{\pi}{4}$ with no vertical shift.

Solution: We use the general form $y = A\cos(Bx - C) + D$.

1. Amplitude is 4: $A = 4$ (we will assume a positive leading coefficient).
2. No vertical shift: $D = 0$.
3. Period is $\pi$: We know $\frac{2\pi}{B} = \text{Period}$. $\frac{2\pi}{B} = \pi \implies B = 2$
4. Phase shift is $\frac{\pi}{4}$ right: We know $\frac{C}{B} = \text{Phase Shift}$. $\frac{C}{2} = \frac{\pi}{4} \implies C = \frac{2\pi}{4} = \frac{\pi}{2}$

Substitute these values back into the general form: $y = 4\cos(2x - \frac{\pi}{2})$