AP PRECALCULUS — UNIT 3A · TRIGONOMETRIC & POLAR FUNCTIONS
3.6A — Sinusoidal Function Transformations
Notes — Vertical Scaling, Shift, and Reflection
💡 Learning Objectives (3.6.A Part 1)
By the end of this lesson you will be able to:
- Apply vertical stretches, compressions, and reflections to sin(x) and cos(x)
- Apply vertical shifts to sinusoidal functions
- Combine multiple vertical transformations in the correct order
- Predict the effect on amplitude, midline, maximum, and minimum values
1. Vertical Scaling by a
Multiplying the output by a stretches or compresses the wave vertically about the midline y = 0. Starting from y = sin(x):
- y = 3 sin(x): amplitude 3, max 3, min −3
- y = (1/2) sin(x): amplitude 1/2, max 1/2, min −1/2
- y = −2 sin(x): amplitude 2, max 2, min −2, but FLIPPED (starts going down)
In general, y = a · sin(x) has amplitude |a|. The sign of a determines whether the wave is flipped.
2. Vertical Shift by d
Adding d to the output moves the whole wave up (or down if d < 0). Starting from y = cos(x):
- y = cos(x) + 4: midline is now y = 4; max 5, min 3
- y = cos(x) − 2: midline is now y = −2; max −1, min −3
The amplitude does NOT change — only where the wave is centered.
3. Combining a and d
The general form y = a · sin(x) + d (or cosine) combines both:
- Amplitude: |a|
- Midline: y = d
- Maximum: d + |a|
- Minimum: d − |a|
📘 Example — Describe y = −4 cos(x) + 7
- a = −4, so amplitude = 4 and the curve is REFLECTED (starts at minimum instead of maximum)
- d = 7, so the midline is y = 7
- Max: 7 + 4 = 11. Min: 7 − 4 = 3
- At x = 0: y = −4(1) + 7 = 3 (the minimum — correct for reflected cosine)
4. Order of Vertical Transformations
When y = a · sin(x) + d, think ‘scale FIRST, shift SECOND.’ In other words, apply the multiplication by a before adding d. Reversing the order changes the graph.
⚠️ Common mistake
The expressions 2 sin(x) + 3 and 2(sin(x) + 3) are NOT the same. The first has midline 3 and amplitude 2; the second has midline 6 and amplitude 2. Read the parentheses carefully.
5. Reflection Across the x-axis
Multiplying by a negative number includes a reflection about the midline (here the x-axis). Crucially:
- y = −sin(x) = sin(−x) (by odd symmetry). A negative vertical flip on sine is the same as a horizontal flip.
- y = −cos(x) is NOT the same as cos(−x). Cosine is even, so cos(−x) = cos(x). To flip cosine, you must reflect VERTICALLY.
Bottom line: a negative ‘a’ always flips the wave vertically, regardless of whether it's sine or cosine. The symmetry observation above is useful but can be misleading.
6. Summary of Effects on Key Features
Transformation | Effect on amplitude | Effect on midline | Effect on max/min |
|---|---|---|---|
Multiply by a > 0 | scales by a | no change | scales by a |
Multiply by a < 0 | scales by |a| | no change | swaps: old max becomes new min after shift |
Add d | no change | shifts by d | both shift by d |
7. Working Backwards
Suppose you see a sinusoidal graph with max 13 and min 1 that behaves like a flipped cosine (starts at the MINIMUM at x = 0).
- Midline: d = (13 + 1)/2 = 7
- Amplitude: |a| = (13 − 1)/2 = 6
- Starts at minimum at x = 0 ⇒ use cosine with NEGATIVE a: a = −6
- Model: y = −6 cos(x) + 7
8. Summary
- Multiplying by a changes the amplitude to |a|; negative a flips the wave
- Adding d shifts the midline to y = d without changing amplitude
- Max = d + |a|, Min = d − |a|
- Apply scaling before shifting — parentheses matter
- Negative a on cosine really does flip; don't rely on cosine's even symmetry