Graphing Sine and Cosine Functions: Amplitude, Period, Phase Shift, and Vertical Shift

Trigonometric Functions and the Unit Circle

Definition of the Sine Function

The sine function is fundamentally defined using the unit circle. For an angle θ in standard position, sin(θ) is the y-coordinate of the point on the unit circle corresponding to that angle.

• Unit Circle: A circle with radius 1 centered at the origin (0,0) in the coordinate plane.

• As the angle θ increases, the point moves counterclockwise around the unit circle, and the y-coordinate traces out the sine function.

• The graph of y = sin(x) is a smooth, continuous wave that repeats every 2π units.

Example: At θ = 0, sin(0) = 0; at θ = π/2, sin(π/2) = 1; at θ = π, sin(π) = 0; at θ = 3π/2, sin(3π/2) = -1; at θ = 2π, sin(2π) = 0.

Reference Angles and Key Values

Using angles from -2π to 2π, the sine function takes on specific values at commonly used reference angles.

Angle (θ)	sin(θ)
0	0
π/6	1/2
π/4	\frac{\sqrt{2}}{2}
π/3	\frac{\sqrt{3}}{2}
π/2	1
π	0
3π/2	-1
2π	0

Properties of the Sine Function

Periodicity and Amplitude

• Period: The sine function repeats every $2\pi$ units. That is, $\sin(x + 2\pi) = \sin(x)$ for all x.

• Amplitude: The amplitude of $y = \sin(x)$ is 1. Amplitude is the maximum vertical distance from the midline (center) of the graph.

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• Odd Function: $\sin(-x) = -\sin(x)$, so the graph is symmetric about the origin.

Summary Table: The Basic Sine Function

Property	y = sin(x)
Domain	$(-\infty, \infty)$
Range	$[-1, 1]$
Amplitude	1
Period	$2\pi$
Graphing Criteria	5 key points per period: 2 maxima, 2 minima, 3 x-intercepts

Transformations of the Sine Function

General Form

The most general form of the sine function is:

$y = a \sin(bx + c) + d$

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Phase Shift: $-\frac{c}{b}$

• Vertical Shift: $d$

Effects of Parameters

• a (Amplitude): Stretches or compresses the graph vertically. If a is negative, the graph is reflected over the x-axis.

• b (Period): Stretches or compresses the graph horizontally. The period is $2\pi/|b|$.

• c (Phase Shift): Shifts the graph left or right by $-c/b$ units.

• d (Vertical Shift): Moves the graph up or down by d units.

Examples of Sine Transformations

• Example 1: $y = 5\sin(x)$ Amplitude = 5, Period = $2\pi$, no phase or vertical shift.

• Example 2: $y = -3\sin(x)$ Amplitude = 3, reflected over x-axis, Period = $2\pi$.

• Example 3: $y = \sin(x/3)$ Amplitude = 1, Period = $6\pi$.

• Example 4: $y = \sin(x-\pi)$ Phase shift right by $\pi$ units.

• Example 5: $y = \sin(4x)$ Period = $\frac{2\pi}{4} = \frac{\pi}{2}$.

• Example 6: $y = \sin(x) + 7$ Vertical shift up 7 units.

• Example 7: $y = \sin(x) - 5$ Vertical shift down 5 units.

• Example 8: $y = 2\sin(3x - \pi) + 1$ Amplitude = 2, Period = $\frac{2\pi}{3}$, Phase shift right by $\frac{\pi}{3}$, Vertical shift up 1.

The Cosine Function

Definition and Properties

The cosine function, $y = \cos(x)$, is also defined using the unit circle. For an angle θ, cos(θ) is the x-coordinate of the point on the unit circle at that angle.

• Period: $2\pi$

• Amplitude: 1

• Domain: $(-\infty, \infty)$

• Range: $[-1, 1]$

• Even Function: $\cos(-x) = \cos(x)$, so the graph is symmetric about the y-axis.

General Form of the Cosine Function

$y = a \cos(bx + c) + d$

• Amplitude: $|a|$

• Period: $\frac{2\pi}{|b|}$

• Phase Shift: $-\frac{c}{b}$

• Vertical Shift: $d$

Examples of Cosine Transformations

• Example 1: $y = 5\cos(x-\pi) - 3$ Amplitude = 5, Period = $2\pi$, Phase shift right by $\pi$, Vertical shift down 3.

• Example 2: $y = \cos\left(-\frac{\pi}{2}x\right) + 4$ Amplitude = 1, Period = $\frac{2\pi}{| -\frac{\pi}{2} |} = \frac{4}{1}$, Phase shift = 0, Vertical shift up 4.

Summary Table: The General Sine and Cosine Functions

Property	y = a sin(bx + c) + d	y = a cos(bx + c) + d
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$[d - |a|, d + |a|]$	$[d - |a|, d + |a|]$
Amplitude	$|a|$	$|a|$
Period	$\frac{2\pi}{|b|}$	$\frac{2\pi}{|b|}$
Phase Shift	$-\frac{c}{b}$	$-\frac{c}{b}$
Vertical Shift	$d$	$d$

Graphing Criteria and Steps

1. Identify amplitude, period, phase shift, and vertical shift from the equation.

2. Determine the interval for one period of the function.

3. Divide the period into four equal parts to find key points (max, min, intercepts).

4. Apply vertical and phase shifts as needed.

5. Plot the five key points and sketch the sinusoidal curve.

Additional info:

• For more complex transformations, use inequalities to determine the new interval for one period after phase and period changes.

• Applications of these functions (such as modeling periodic phenomena) are covered in a different set of notes.