Graphs of Trig Functions
Understand the shape, period, amplitude, and phase shift of sine and cosine graphs β and learn to sketch and interpret transformed trig functions of the form y = A sin(Bx + C) + D.
Sine and cosine waves are the mathematical shape of sound, light, radio signals, ocean waves, AC electricity, and the motion of a pendulum. When an audio engineer adjusts EQ, they are changing the amplitude of specific sine wave frequencies. When a doctor reads an ECG, they are interpreting a periodic wave. Understanding trig graphs β and how the parameters A, B, C, and D shift and stretch them β is essential for any field that deals with oscillation or periodic phenomena.
- Identify the amplitude, period, and midline of a sine or cosine graph
- Apply transformations: vertical stretch (A), frequency (B), phase shift (C), vertical shift (D)
- Sketch \(y = A\sin(Bx + C) + D\) from its parameters
- Interpret a real-world periodic graph and extract its equation
- A: Amplitude β stretches or compresses vertically (and reflects if negative).
- B: Controls period: \(T = \frac{2\pi}{B}\).
- C: Creates a phase (horizontal) shift of \(-C/B\).
- D: Shifts the midline up or down by D units.
The same formula applies to cosine: \(y = A\cos(Bx + C) + D\). The only difference is that cosine starts at a peak (not the midline) when there is no phase shift.
Key Features of y = sin(x) vs y = cos(x)
| Feature | y = sin(x) | y = cos(x) |
|---|---|---|
| Amplitude | 1 | 1 |
| Period | 360Β° (2Ο) | 360Β° (2Ο) |
| Starts at (0, ?) | 0 (midline) | 1 (peak) |
| First peak at | 90Β° (Ο/2) | 0Β° |
| Range | [β1, 1] | [β1, 1] |
For \(y = 3\sin(2x)\), identify amplitude, period, and midline.
- Amplitude: \(|A| = 3\). The graph peaks at 3 and troughs at β3.
- Period: \(T = 360Β°/2 = 180Β°\). One full cycle completes in 180Β°.
- Midline: \(D = 0\). The wave oscillates around the x-axis.
For \(y = -2\cos(3x - 90Β°) + 4\), find amplitude, period, phase shift, and midline.
- Amplitude: \(|A| = |-2| = 2\). Reflected (negative A flips the graph).
- Period: \(T = 360Β°/3 = 120Β°\).
- Phase shift: \(-C/B = -(-90Β°)/3 = +30Β°\) to the right.
- Midline: \(y = 4\). Graph oscillates between \(4-2=2\) and \(4+2=6\).
A musical note A4 (concert A) has a frequency of 440 Hz. Its sound wave is \(y = P\sin(2\pi \times 440 \times t)\) where P is peak pressure. What is the period (time for one cycle)?
\[ T = \frac{1}{440 \text{ Hz}} \approx \mathbf{0.00227 \text{ seconds}} \approx 2.27 \text{ ms} \]That's how fast the air pressure oscillates to produce the note A. A higher pitch has a shorter period (higher frequency, higher B).
- For \(y = 5\sin(x/2)\), what is the amplitude and period?
- A sine wave completes 3 full cycles in 360Β°. What is B?
- Describe how \(y = \sin(x) + 2\) differs from \(y = \sin(x)\).
βΆ Show Answers
- Amplitude = 5; Period = \(360Β°/(1/2) = \mathbf{720Β°}\).
- Period = 360Β°/3 = 120Β°, so \(T = 360Β°/B\) β \(B = \mathbf{3}\).
- The midline shifts from y = 0 to y = 2. Every y-value increases by 2; peaks at 3, troughs at 1.
- Phase shift sign error: For \(\sin(Bx + C)\), the phase shift is \(-C/B\) β note the negative sign. \(\sin(x + 90Β°)\) shifts LEFT 90Β°, not right.
- Confusing amplitude with range: Amplitude is |A| β always positive. Range is [Dβ|A|, D+|A|]. The midline D shifts the range; amplitude determines the half-height.
- Forgetting B affects period, not frequency directly: Period = 360Β°/B. Doubling B halves the period (doubles the frequency). Don't confuse B with frequency directly.
- A = amplitude, B controls period (\(T = 360Β°/B\)), C causes phase shift (\(-C/B\)), D is the midline.
- Sine starts at the midline; cosine starts at the peak.
- A negative A reflects the graph vertically.
- Frequency and period are reciprocals: \(f = 1/T\).
Every sound is a sum of sine waves β this is the core insight of Fourier analysis. Audio engineers manipulate amplitude (volume), frequency (pitch), and phase (timing) of sine components to mix, equalize, and master audio. Digital audio effects like reverb, chorus, and flanger all work by adding delayed or frequency-shifted copies of the original sine wave components. Understanding trig graphs is the first step toward understanding signal processing at any level.
Calculator Connection
The Sine Wave Generator plots \(y = A\sin(Bx + C) + D\) interactively. The Wave Transformation Calculator lets you adjust parameters and see how amplitude, period, phase, and vertical shift change the graph in real time.
Try it with the Calculator
Apply what you've learned with these tools.
Graphs of Trig Functions β Quiz
5 questions per attempt Β· Intermediate Β· Pass at 70%
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