Graphs of Sine and Cosine

By the end of this lesson, you will be able to graph $y = a\sin(bx - c) + d$ and $y = a\cos(bx - c) + d$, identify amplitude, period, phase shift, and vertical shift, and write equations from graphs.

The Parent Graphs

$y = \sin x$:

• Range: $[-1, 1]$; Period: $2\pi$
• Passes through $(0,0)$, $(π/2, 1)$, $(\pi, 0)$, $(3\pi/2, -1)$, $(2\pi, 0)$

$y = \cos x$:

• Range: $[-1, 1]$; Period: $2\pi$
• Passes through $(0,1)$, $(\pi/2, 0)$, $(\pi, -1)$, $(3\pi/2, 0)$, $(2\pi, 1)$

Both are continuous, smooth, periodic waves.

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

Parameter	Effect
$	a
$a < 0$	Reflection over $x$-axis
$b$	Period = $2\pi/
$c/b$	Phase shift (horizontal shift right)
$d$	Vertical shift (midline at $y = d$)

Midline: $y = d$ (the horizontal axis of the wave)

Maximum: $d + |a|$; Minimum: $d - |a|$

Period Formula

$$\text{Period} = \frac{2\pi}{|b|}$$

If $b > 1$: compressed (faster oscillation). If $0 < b < 1$: stretched (slower).

Phase Shift

$$\text{Phase shift} = \frac{c}{b}$$

Positive phase shift → graph shifts right. Negative → shifts left.

Important: write in the form $b(x - c/b)$ to see the phase shift clearly.

Example 1 — Full Analysis

$y = 3\sin(2x - \pi) + 1$

$|a| = 3$ → amplitude 3

$b = 2$ → period $= 2\pi/2 = \pi$

Phase shift $= \pi/2$ (right)

$d = 1$ → midline $y = 1$

Maximum: $1 + 3 = 4$; Minimum: $1 - 3 = -2$

Key point: first cycle starts at $x = \pi/2$ (where $2x - \pi = 0$) and ends at $x = \pi/2 + \pi = 3\pi/2$.

Example 2 — Cosine with Reflection

$y = -2\cos(x/2) + 3$

Amplitude: 2 (reflected over midline)

Period: $2\pi/(1/2) = 4\pi$

Phase shift: 0 (no $c$ term)

Midline: $y = 3$; Max: 5; Min: 1

Since reflected: starts at minimum $(0, 1)$ instead of maximum.

Example 3 — Writing the Equation from a Graph