Section 6.1

Graphs of Sine & Cosine

Trig graph questions show up as 2–3 problems on Exam 3. You'll need to go both directions: reading parameters from an equation, and writing an equation from a description. These graphs also help you visualize solutions to trig equations in Section 7.5.

⚡ Quick Summary

Sine and cosine make wave patterns — think of a heartbeat on a monitor, ocean tides, or a sound wave. The basic $y = \sin(x)$ wave bobs smoothly between $-1$ and $1$ , repeating every $2\pi$ units. Cosine is the exact same wave, just shifted sideways a little. Every fancy sinusoidal function you'll see is just one of these waves that's been stretched, squished, shifted, or flipped.

General Sinusoidal Form

y = A\sin(Bx - C) + D \quad \text{or} \quad y = A\cos(Bx - C) + D

A, B, C, D are real constants with B > 0

Amplitude

\text{Amplitude} = |A|

The height from the midline to a peak. If A is negative, the graph is reflected vertically.

📋 Before You Start

Tap any item if you need a refresher:

Plain-English version

The general equation $y = A\sin(Bx - C) + D$ has four "knobs" you can turn. A controls how tall the wave is (the amplitude — the height from the middle to a peak). B controls how wide each cycle is (the period $= \frac{2\pi}{|B|}$ ). C slides the wave left or right (the phase shift $= \frac{C}{B}$ ). D moves the whole wave up or down (the vertical shift, or midline). Think of it like an A-B-C-D checklist.

The biggest trap is the phase shift. It's $\frac{C}{B}$ , not just $C$ . For example, in $y = \sin(2x - \pi)$ , the shift is $\frac{\pi}{2}$ , not $\pi$ . Also, a negative $A$ flips the wave upside down, but the amplitude is still the positive number $|A|$ — amplitude measures height, so it's never negative.

If the exam gives you an equation, just read off A, B, C, D and calculate the four parameters. If they describe a wave in words ("amplitude 3, period $\pi$ "), work backwards — find B from the period formula and C from the phase shift. The range of the wave is always $[D - |A|,\, D + |A|]$ .

Interactive Diagram

Key Formulas

General Sinusoidal Form

y = A\sin(Bx - C) + D \quad \text{or} \quad y = A\cos(Bx - C) + D

A, B, C, D are real constants with B > 0

Amplitude

\text{Amplitude} = |A|

The height from the midline to a peak. If A is negative, the graph is reflected vertically.

Period

\text{Period} = \frac{2\pi}{|B|}

The horizontal length of one full cycle

Phase Shift

\text{Phase Shift} = \frac{C}{B}

Positive means shift right, negative means shift left

Vertical Shift

\text{Vertical Shift} = D

The midline of the graph is y = D

Key Takeaways

✓General form: $y = A\sin(Bx - C) + D$ — amplitude $|A|$ , period $\frac{2\pi}{|B|}$ , phase shift $\frac{C}{B}$ , vertical shift $D$
✓Phase shift is $\frac{C}{B}$ , not just $C$ — this is the most common mistake
✓Negative $A$ flips the graph vertically, but amplitude is always $|A|$ (positive)
✓Range of a sinusoidal function: $[D - |A|,\, D + |A|]$

⚠️ Common Mistakes

✗Confusing phase shift with $C$ — the phase shift is $\frac{C}{B}$ , not just $C$ (e.g., in $y = \sin(2x - \pi)$ , the shift is $\frac{\pi}{2}$ , not $\pi$ )
✗Saying amplitude is negative — amplitude is always $|A|$ , a positive number; a negative $A$ flips the graph but doesn't make amplitude negative
✗Forgetting to factor out $B$ before reading the phase shift — rewrite the argument as $B(x - \frac{C}{B})$ to see the shift clearly
✗Mixing up range and period — range is the vertical interval $[D - |A|,\, D + |A|]$ ; period is the horizontal length $\frac{2\pi}{|B|}$

Next: Section 6.2 Builds on: 5.2

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5.4 Right Triangle Trigonometry

6.2 Graphs of Other Trig Functions