Graphing Sine and Cosine Functions

Learning Targets

• Find and interpret the following for any sine or cosine function:

  • Amplitude (height of the wave)

  • Period (length of one cycle)

  • Phase/Horizontal Shift (left–right displacement of the graph)

  • Vertical Shift/Displacement (up–down movement of the midline)

• Sketch accurate graphs of $y = \sin x$ and $y = \cos x$ and any transformed variants.

General Form of the Functions

• Sine: $y = a\,\sin\bigl(b\,(x - h)\bigr) + k$

• Cosine: $y = a\,\cos\bigl(b\,(x - h)\bigr) + k$

Meaning of Each Parameter

• Amplitude

  • $|a|$

  • Distance from the midline to a peak or trough.

  • Negative $a$ reflects the graph vertically (flips top ↔ bottom), but the amplitude is still positive.

• Period

  • $\dfrac{2\pi}{|b|}$ for both sine and cosine.

  • The graph repeats every one full period.

• Phase (Horizontal) Shift

  • $h$

  • If the expression is $(x - h)$, the wave shifts $h$ units right; if $(x + h)$ it shifts $h$ units left.

• Vertical Shift

  • $k$

  • Moves the entire graph up ($k>0$) or down ($k<0$).

Five-Key-Point Strategy for Graphing

1. Determine amplitude, period, phase shift, and vertical shift.

2. Compute the step size (period ÷ 4). This partitions one cycle into the five classic points:

   • Start (midline)

   • Quarter-period point (peak for sine, still peak for cosine unless flipped)

   • Half-period (midline again)

   • Three-quarter-period (trough)

   • Full period (back to starting midline)

3. Apply horizontal and vertical shifts to every x- and y-coordinate.

4. Plot additional cycles if needed by repeating the pattern.

5. Connect points with a smooth, continuous wave.

Example 1 $y = \sin x$

• Amplitude: $1$

• Period: $2\pi$

• Phase Shift: $0$

• Vertical Shift: $0$

• Key points (x, y):

  • $(0,0)$

  • $\bigl(\frac{\pi}{2},1\bigr)$

  • $(\pi,0)$

  • $\bigl(\frac{3\pi}{2},-1\bigr)$

  • $(2\pi,0)$

• Highest value $=1$, lowest value $=-1$.

Example 2 $y = -\cos\bigl(2(x - \tfrac{\pi}{4})\bigr)$

• Amplitude: $1$ (negative sign means vertical reflection)

• Period: $\dfrac{2\pi}{2}=\pi$

• Phase Shift: $\tfrac{\pi}{4}$ units right

• Vertical Shift: $0$

• Step size: $\frac{\pi}{4}$

• Reference five points (before reflection): $0,\tfrac{\pi}{4},\tfrac{\pi}{2},\tfrac{3\pi}{4},\pi$ then apply shift & reflection.

  • Because of the “−” the usual cosine peak becomes a trough.

  • Start point at $x = \tfrac{\pi}{4}$ is now $y = -1$ (minimum).

Example 3 $y = 3\sin x - 2$

• Amplitude: $3$

• Period: $2\pi$

• Phase Shift: $0$

• Vertical Shift: $-2$ (midline $y=-2$)

• Peak: $-2 + 3 = 1$

• Trough: $-2 - 3 = -5$

• Key midline points: $(0,-2),\ (\pi,-2),\ (2\pi,-2)$

• Peaks & troughs $\bigl(\tfrac{\pi}{2},1\bigr)$ and $\bigl(\tfrac{3\pi}{2},-5\bigr)$.

Example 4 $y = \cos\bigl(\tfrac{\pi}{4}(x - 2)\bigr) + 1$

• Amplitude: $1$

• Period: $\dfrac{2\pi}{\pi/4} = 8$

• Phase Shift: $2$ units right

• Vertical Shift: $+1$ (midline $y=1$)

• Step size: $\tfrac{8}{4}=2$

• Key x-coordinates (one cycle): $2,4,6,8,10$

  • Start at $x=2$ on a peak because cosine normally starts at $1$. With no reflection, $y=1+1=2$.

  • Continue down to mid, trough, mid, and back to peak.

Additional Tips & Common Errors

• Always factor the coefficient of $x$ before reading the phase shift. $\cos(2x-\pi)$ is really $\cos\bigl(2(x-\tfrac{\pi}{2})\bigr)$, so the shift is $\tfrac{\pi}{2}$, not $\pi$.

• Negative amplitude produces a vertical flip; do not change the numerical value of the amplitude.

• To locate successive cycles, just keep adding or subtracting the period to every x-coordinate.

• When mixing degrees and radians, be consistent; these notes use radians throughout.

Ethical / Practical Connection

• Clear visualization of trigonometric waves is foundational for physics (sound, light), engineering (signal processing), and even economics (cyclical trends). Mis-reading a phase shift or period can translate into real-world design errors—e.g., antennas cut to the wrong wavelength.

• Good communication matters: if any step is unclear, ask immediately (as your instructor noted: “don’t hesitate to message me”).