Sine Function Graphs

Master the sine function from its basic form y = sin x to complex transformations y = A sin(ωx + φ) + b. Understand domain, range, periodicity, symmetry properties, monotonic intervals, and systematic graph transformations.

Sine Function Graph: y = sin x

Visual representation of the basic sine function

Graph Features: This extended view shows the sine function over two complete cycles from -2π to 2π. The function passes through zero at -2π, -π, 0, π, and 2π; reaches maximum value of 1 at -3π/2 and π/2; and reaches minimum value of -1 at -π/2 and 3π/2.

Basic Sine Function Properties: y = sin x

Fundamental characteristics and values

Domain

ℝ (all real numbers)

Sine function is defined for all real inputs

Range

[-1, 1]

Output values are bounded between -1 and 1

Period

2π

Function repeats every 2π units

Parity

Odd function

f(-x) = -sin x = -f(x)

Zeros

x = kπ (k ∈ ℤ)

Function equals zero at integer multiples of π

Maximum

x = π/2 + 2kπ

Reaches maximum value of 1

Minimum

x = 3π/2 + 2kπ

Reaches minimum value of -1

Monotonic Intervals

Where the sine function increases and decreases

Increasing Intervals

[2kπ - π/2, 2kπ + π/2]

Where k ∈ ℤ. Function rises from -1 to 1.

Decreasing Intervals

[2kπ + π/2, 2kπ + 3π/2]

Where k ∈ ℤ. Function falls from 1 to -1.

Symmetry Properties

Lines and points of symmetry for y = sin x

Symmetry Centers

(kπ, 0) where k ∈ ℤ

Points of 180° rotational symmetry at zeros.

Symmetry Axes

x = kπ + π/2 where k ∈ ℤ

Vertical lines through maximum and minimum points.

Composite Sine Function: y = A sin(ωx + φ)

Advanced properties with amplitude, frequency, and phase parameters

Parameter Effects (A >0, ω >0)

A (Amplitude): Controls vertical stretch and range [-A, A]
ω (Angular frequency): Controls period T = 2π/ω and horizontal compression
φ (Phase shift): Controls horizontal translation of the wave

Period

T = 2π/ω

Period inversely proportional to frequency ω

Range

[-A, A]

Amplitude A determines the vertical stretch

Maximum Points

ωx + φ = π/2 + 2kπ

Solve for x to find maximum locations

Minimum Points

ωx + φ = 3π/2 + 2kπ

Solve for x to find minimum locations

Symmetry Axis

x = (π/2 + kπ - φ)/ω

Vertical lines of symmetry through extrema

Symmetry Center

((kπ - φ)/ω, 0)

Points of rotational symmetry through zeros

Calculation Methods for y = A sin(ωx + φ)

Step-by-step calculation processes for key properties

1. Finding Symmetry Axes (Maximum/Minimum Points)

Step 1: Set up the condition for extrema

ωx + φ = π/2 + kπ (where k ∈ ℤ)

This condition captures both maximum (k even) and minimum (k odd) points.

Step 2: Solve for x

x = (π/2 + kπ - φ)/ω

These x-values are the symmetry axes of the function.

Example: For y = 2sin(3x + π/4)

A = 2, ω = 3, φ = π/4

3x + π/4 = π/2 + kπ

3x = π/4 + kπ

x = π/12 + kπ/3

Symmetry axes: x = π/12, 5π/12, 3π/4, ...

2. Finding Symmetry Centers (Zero Points)

Step 1: Set up the condition for zeros

ωx + φ = kπ (where k ∈ ℤ)

The sine function equals zero at multiples of π.

Step 2: Solve for x

x = (kπ - φ)/ω

Symmetry centers are at ((kπ - φ)/ω, 0).

Example: For y = 2sin(3x + π/4)

3x + π/4 = kπ

x = (kπ - π/4)/3 = π(4k - 1)/12

Centers: (-π/12, 0), (π/4, 0), (7π/12, 0), ...

3. Finding Monotonic Intervals

Step 1: Increasing intervals condition

ωx + φ ∈ [-π/2 + 2kπ, π/2 + 2kπ]

Where the derivative sin'(ωx + φ) = ω cos(ωx + φ) > 0

Step 2: Solve the inequality

-π/2 + 2kπ ≤ ωx + φ ≤ π/2 + 2kπ

-π/2 + 2kπ - φ ≤ ωx ≤ π/2 + 2kπ - φ

(-π/2 + 2kπ - φ)/ω ≤ x ≤ (π/2 + 2kπ - φ)/ω

Step 3: Decreasing intervals condition

ωx + φ ∈ [π/2 + 2kπ, 3π/2 + 2kπ]

Solution: x ∈ [(π/2 + 2kπ - φ)/ω, (3π/2 + 2kπ - φ)/ω]

Example: For y = 2sin(3x + π/4)

Increasing:

x ∈ [(-3π/4 + 2kπ)/3, (π/4 + 2kπ)/3] = [-π/4 + 2kπ/3, π/12 + 2kπ/3]

Decreasing:

x ∈ [(π/4 + 2kπ)/3, (5π/4 + 2kπ)/3] = [π/12 + 2kπ/3, 5π/12 + 2kπ/3]

4. Finding Maximum and Minimum Values

Step 1: Maximum value calculation

Condition: ωx + φ = π/2 + 2kπ (where sin = 1)

x = (π/2 + 2kπ - φ)/ω

Maximum value: y = A·1 = A

Step 2: Minimum value calculation

Condition: ωx + φ = 3π/2 + 2kπ (where sin = -1)

x = (3π/2 + 2kπ - φ)/ω

Minimum value: y = A·(-1) = -A

Example: For y = 2sin(3x + π/4)

Maximum: y = 2 at x = π/12 + 2kπ/3

Minimum: y = -2 at x = 5π/12 + 2kπ/3

Graph Transformations

Step-by-step transformation from y = sin x to y = A sin(ωx + φ) + b

1. Phase Shift

Transformation:

sin x → sin(x + φ)

Effect:

Horizontal shift left (φ > 0) or right (φ < 0) by |φ| units

Visual Impact:

Changes the starting position of the wave

2. Frequency/Period Change

Transformation:

sin(x + φ) → sin(ωx + φ)

Effect:

Horizontal compression by factor 1/ω, period becomes 2π/ω

Visual Impact:

Changes how quickly the wave oscillates

3. Amplitude Change

Transformation:

sin(ωx + φ) → A sin(ωx + φ)

Effect:

Vertical stretch by factor A, range becomes [-A, A]

Visual Impact:

Changes the height of the wave peaks and valleys

4. Vertical Shift

Transformation:

A sin(ωx + φ) → A sin(ωx + φ) + b

Effect:

Vertical shift up (b > 0) or down (b < 0) by |b| units

Visual Impact:

Moves the entire wave up or down, changing the midline

Key Mathematical Insights

Essential understanding for mastering sine function graphs

Order of Transformations Matters

When transforming y = sin x to y = A sin(ωx + φ) + b, the order affects intermediate results. The standard order preserves mathematical clarity: phase → frequency → amplitude → vertical shift.

Symmetry and Periodicity

The sine function's odd symmetry (f(-x) = -f(x)) and 2π-periodicity are preserved under transformations, though the specific symmetry points and period change according to the transformation parameters.

Practical Applications

Sine functions model oscillatory phenomena in physics (waves, vibrations), engineering (AC circuits), and natural sciences (seasonal variations, biorhythms). Understanding transformations enables modeling real-world periodic behavior.