The three main trigonometric functions have distinct graphs:

Sine Function: $y = \sin(x)$

• Domain: All real numbers $(-\infty, \infty)$
• Range: $[-1, 1]$
• Period: $2\pi$
• Amplitude: 1
• Key points: $(0, 0)$, $(\frac{\pi}{2}, 1)$, $(\pi, 0)$, $(\frac{3\pi}{2}, -1)$, $(2\pi, 0)$
• Zeros: $x = n\pi$ where $n$ is an integer

Cosine Function: $y = \cos(x)$

• Domain: All real numbers $(-\infty, \infty)$
• Range: $[-1, 1]$
• Period: $2\pi$
• Amplitude: 1
• Key points: $(0, 1)$, $(\frac{\pi}{2}, 0)$, $(\pi, -1)$, $(\frac{3\pi}{2}, 0)$, $(2\pi, 1)$
• Zeros: $x = \frac{\pi}{2} + n\pi$ where $n$ is an integer

Tangent Function: $y = \tan(x)$

• Domain: All real numbers except $x = \frac{\pi}{2} + n\pi$ where $n$ is an integer
• Range: All real numbers $(-\infty, \infty)$
• Period: $\pi$
• Vertical asymptotes: $x = \frac{\pi}{2} + n\pi$
• Key points: $(0, 0)$, $(\frac{\pi}{4}, 1)$, $(-\frac{\pi }{4},$
• Zeros: $x = n\pi$ where $n$ is an integer

General Form and Transformations

The general form of a sinusoidal function is:

$y = A\sin(B(x - C)) + D \quad \text{or} \quad y = A\cos(B(x - C)) + D$

Where:

• $|A|$ = Amplitude (vertical stretch/compression)
  • If $A < 0$, the graph is reflected over the x-axis
• $B$ affects the Period: Period = $\frac{2\pi}{|B|}$
  • If $B > 1$, horizontal compression (shorter period)
  • If $0 < B < 1$, horizontal stretch (longer period)
• $C$ = Phase shift (horizontal translation)
  • If $C > 0$, shift right
  • If $C < 0$, shift left
• $D$ = Vertical shift (midline)
  • The midline is $y = D$

For Tangent Functions

$y = A\tan(B(x - C)) + D$

• Period = $\frac{\pi}{|B|}$
• Vertical asymptotes at $x = C + \frac{\pi}{2B} + \frac{n\pi}{B}$ where $n$ is an integer

Finding Key Features from Equations

Given $y = A\sin(B(x - C)) + D$:

1. Amplitude: $|A|$
2. Period: $\frac{2\pi}{|B|}$
3. Phase Shift: $C$
4. Vertical Shift/Midline: $y = D$
5. Maximum value: $D + |A|$
6. Minimum value: $D - |A|$

Graphing Strategy

1. Identify $A$, $B$, $C$, and $D$
2. Draw the midline at $y = D$
3. Mark the amplitude (max at $D + |A|$, min at $D - |A|$)
4. Find the period and mark one complete cycle
5. Apply phase shift (start at $x = C$)
6. Plot key points within one period
7. Sketch the curve and extend if needed

Common Patterns

Cosecant and Secant

• $\csc(x) = \frac{1}{\sin(x)}$: Has vertical asymptotes where $\sin(x) = 0$
• $\sec(x) = \frac{1}{\cos(x)}$: Has vertical asymptotes where $\cos(x) = 0$

Cotangent

• $\cot(x) = \frac{1}{\tan(x)}$: Period is $\pi$, asymptotes where $\tan(x) = 0$

• $A = 2$
• $B = 1$
• $C = 0$
• $D = 1$

Features:

1. Amplitude: $|A| = |2| = 2$
2. Period: $\frac{2\pi}{|B|} = \frac{2\pi}{1} = 2\pi$
3. Phase Shift: $C = 0$ (no horizontal shift)
4. Midline: $y = D = 1$
5. Maximum: $1 + 2 = 3$
6. Minimum: $1 - 2 = -1$

Key Points (one period from $x = 0$ to $x = 2\pi$):

• Start: $(0, 1)$ (midline)
• Max: $(\frac{\pi}{2}, 3)$
• Midline: $(\pi, 1)$
• Min: $(\frac{3\pi}{2}, -1)$
• End: $(2\pi, 1)$ (midline)

The graph oscillates between $y = -1$ and $y = 3$, centered on the midline $y = 1$.

a) Find the amplitude. b) Find the period. c) Find the phase shift. d) Find the vertical shift (midline).

a) Determine the amplitude, period, phase shift, and midline. b) Find the maximum and minimum values of $g(x)$. c) Find the first three $x$-intercepts for $x \geq 0$.