Trigonometric Graphs

Basic Sine Function

y = sin(x)

Key features:

• Domain: All real numbers
• Range: [-1, 1]
• Period: 2π (repeats every 2π)
• Amplitude: 1
• Zeros: x = 0, π, 2π, 3π, ...

Shape: Wave starting at origin, going up to 1, down through 0 to -1, back to 0

Key points in one period [0, 2π]:

• (0, 0)
• (π/2, 1) — maximum
• (π, 0)
• (3π/2, -1) — minimum
• (2π, 0)

Basic Cosine Function

y = cos(x)

Key features:

• Domain: All real numbers
• Range: [-1, 1]
• Period: 2π
• Amplitude: 1
• Zeros: x = π/2, 3π/2, 5π/2, ...

Shape: Wave starting at maximum (1), similar to sine but shifted

Key points in one period [0, 2π]:

• (0, 1) — maximum
• (π/2, 0)
• (π, -1) — minimum
• (3π/2, 0)
• (2π, 1) — maximum

Relationship: cos(x) = sin(x + π/2)

Basic Tangent Function

y = tan(x)

Key features:

• Domain: All real numbers except π/2 + nπ
• Range: All real numbers
• Period: π (repeats every π, shorter than sin/cos!)
• No amplitude (unbounded)
• Vertical asymptotes at x = π/2, 3π/2, ...
• Zeros: x = 0, π, 2π, ...

Shape: Repeating S-curves between vertical asymptotes

Amplitude

Amplitude: Vertical stretch/compression

Form: y = A·sin(x) or y = A·cos(x)

|A| = amplitude (distance from center to max/min)

• If |A| > 1: vertical stretch
• If 0 < |A| < 1: vertical compression
• If A < 0: reflection over x-axis

Example 1: Amplitude Change

y = 3sin(x)

Amplitude: 3 Range: [-3, 3] Stretches graph vertically by factor of 3

Example 2: Negative Amplitude

y = -2cos(x)

Amplitude: 2 Range: [-2, 2] Reflected and stretched

Period

Period: Horizontal length of one complete cycle

Form: y = sin(Bx) or y = cos(Bx)

Period = 2π/|B|

• If |B| > 1: horizontal compression (faster oscillation)
• If 0 < |B| < 1: horizontal stretch (slower oscillation)

Example 1: Faster Oscillation

y = sin(2x)

B = 2 Period = 2π/2 = π Completes 2 cycles in 2π (instead of 1)

Example 2: Slower Oscillation

y = cos(x/2)

B = 1/2 Period = 2π/(1/2) = 4π Completes 1 cycle in 4π (instead of 2π)

Example 3: Tangent Period

y = tan(2x)

For tangent: Period = π/|B| Period = π/2

Phase Shift (Horizontal Shift)

Phase shift: Horizontal translation left or right

Form: y = sin(x - C) or y = cos(x - C)

Phase shift = C

• C > 0: shift right
• C < 0: shift left

Example 1: Shift Right

y = sin(x - π/2)

Phase shift: π/2 to the right Starts at x = π/2 instead of x = 0

Example 2: Shift Left

y = cos(x + π)

Rewrite: y = cos(x - (-π)) Phase shift: π to the left

Vertical Shift

Vertical shift: Move entire graph up or down

Form: y = sin(x) + D or y = cos(x) + D

Vertical shift = D

New midline: y = D (instead of y = 0)

Example: Vertical Shift

y = sin(x) + 2

Midline: y = 2 Range: [1, 3] (shifted up 2 units) Maximum: 3, Minimum: 1

General Form

y = A·sin(B(x - C)) + D

y = A·cos(B(x - C)) + D

Where:

• |A| = amplitude
• 2π/|B| = period
• C = phase shift
• D = vertical shift

Example: Complete Transformation

y = 2sin(3(x - π/4)) + 1

Amplitude: 2 Period: 2π/3 Phase shift: π/4 right Vertical shift: 1 up Range: [-1, 3]

Graphing Strategy

To graph y = A·sin(B(x - C)) + D:

1. Identify amplitude |A|, period 2π/B, phase shift C, vertical shift D
2. Draw midline at y = D
3. Mark one period starting at x = C
4. Divide period into 4 equal parts
5. Plot 5 key points
6. Connect with smooth curve
7. Extend pattern

Example: Graph y = 3cos(2x) - 1

Step 1: Amplitude = 3, Period = π, Phase shift = 0, Vertical shift = -1

Step 2: Midline at y = -1

Step 3: One period from 0 to π

Step 4: Divide π into 4 parts: 0, π/4, π/2, 3π/4, π

Step 5: Key points (cosine starts at max):

• (0, 2) — max: -1 + 3
• (π/4, -1) — midline
• (π/2, -4) — min: -1 - 3
• (3π/4, -1) — midline
• (π, 2) — max

Tangent Transformations

y = A·tan(B(x - C)) + D

• No amplitude (tangent is unbounded)
• Period = π/|B|
• Vertical asymptotes at x = C + π/(2B) + nπ/B

Example: Transformed Tangent

y = 2tan(x - π/4)

Period: π Phase shift: π/4 right Vertical stretch: factor of 2 Asymptotes shift right π/4

Determining Equation from Graph

From graph, identify:

1. Amplitude (distance from midline to max)
2. Period (horizontal length of one cycle)
3. Phase shift (where cycle starts)
4. Vertical shift (midline location)

Example: Write Equation

Graph shows:

• Max = 5, Min = -1
• One cycle from 0 to π
• Cosine shape

Amplitude: (5 - (-1))/2 = 3 Vertical shift: (5 + (-1))/2 = 2 Period = π, so 2π/B = π → B = 2

Equation: y = 3cos(2x) + 2

Real-World Applications

Sound waves: Amplitude = volume, frequency = pitch

Tides: Periodic rise and fall of sea level

Daylight hours: Seasonal variation

AC electricity: Voltage oscillates sinusoidally

Springs: Oscillating motion

Example: Temperature Model

Daily temperature modeled by:

T(t) = 15cos(π(t - 14)/12) + 20

Where t = hours after midnight

Amplitude: 15° Period: 24 hours Max temp: 35° at t = 14 (2 PM) Min temp: 5° at t = 2 (2 AM) Average: 20°

Common Transformations Summary

y = 2sin(x): Amplitude 2 y = sin(2x): Period π y = sin(x - π): Shift right π y = sin(x) + 2: Shift up 2 y = -sin(x): Reflection over x-axis y = sin(-x): Reflection over y-axis (same as -sin(x))

Practice