Amplitude Period Phase Shift Calculator: How to Read A·sin(Bx-C)+D
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Marko Šinko
Co-Founder & Lead Developer, AI Math Calculator
An amplitude period phase shift calculator exists because every sinusoidal function question — every one — reduces to reading four numbers out of y = A·sin(Bx − C) + D. Amplitude is |A|. Period is 2π/|B|. Phase shift is C/B. The midline sits at y = D. Four formulas, four graph transformations, and the entire “graphing sine and cosine” unit of precalculus is just combinations of them.
Two of the four are traps, though. The phase shift is C divided by B, not C itself, and getting that wrong on y = 3·sin(2x − π/2) + 1 doubles your answer. And a negative A does not mean negative amplitude — no such thing exists. The calculator above parses your equation, prints each formula with your numbers substituted, and graphs the result next to the same wave without its phase shift so you can literally see the slide. This article is the companion reference: what each number does, where the two traps live, how to run the process backwards from a graph, and a table of common cases to check yourself against.
The Four Numbers an Amplitude Period Phase Shift Calculator Reads
Each coefficient owns exactly one transformation, and none of them interfere with each other. Using y = 3·sin(2x − π/2) + 1 as the running example:
| Symbol | What it controls | Formula | Here |
|---|---|---|---|
| A = 3 | Height from midline to peak (vertical stretch) | amplitude = |A| | 3 |
| B = 2 | How fast it repeats (horizontal compression) | period = 2π/|B| | π ≈ 3.1416 |
| C = π/2 | Horizontal slide (left or right) | phase shift = C/B | π/4 right |
| D = 1 | Where the centre line sits (vertical slide) | midline y = D | y = 1 |
Everything else is derived. Maximum = D + |A| = 4. Minimum = D − |A| = −2. The wave completes 1/period ≈ 0.32 cycles per unit of x. Whether the base function is sine or cosine changes none of these numbers — cos(x) is just sin(x) slid left by π/2, one of the identities a trig identity calculator can verify — so the four formulas apply to both.
Why the Phase Shift Is C/B, Not C
Here’s the trap that fills red ink on precalculus exams. In y = 3·sin(2x − π/2) + 1, the π/2 is not the phase shift. A horizontal shift only reads off cleanly when the inside is factored so that x stands alone:
3·sin(2x − π/2) + 1 = 3·sin(2(x − π/4)) + 1
Factor the 2 out of the parentheses and the shift shows its true size: π/4, half of what the unfactored equation seems to say. That’s the entire reason a phase shift calculator divides by B. The rule works in both directions, which is why this calculator offers both textbook forms — type C = π/2 in the standard form, flip the form selector, and watch it become h = π/4 in the factored one. Same function, same graph.
Direction is the second half of the trap. The standard form has a minus sign built in, so sin(2x − π/2) shifts right — the sign you see is already spoken for. Meet sin(2x + π/3) and the shift goes left by π/6, because +π/3 means C = −π/3. The convention stops feeling arbitrary once you watch what subtracting does: in sin(2x − π/2) the inside doesn’t reach 0 until x = π/4, so every feature of the wave arrives late — the graph slides right. Tracing where those inside angles land on a unit circle calculator is the quickest way to see the direction rule for yourself.
Period and B: The Compression Factor
A plain sine wave repeats every 2π radians. Multiplying x by B makes the input race B times faster, so the wave finishes a full cycle in 2π/|B|. Three quick reads: B = 2 gives period π. B = 1/2 gives period 4π — a stretch, not a squeeze. B = π gives period exactly 2, which is why modelling problems love it: y = 5·sin(πx) completes one clean cycle every 2 hours, days, or metres, no π left in the answer.
The absolute value matters more than it looks. sin(−2x) still has period π, not −π — a period is a length, and lengths aren’t signed. And if x is measured in degrees, swap the 2π for 360°: y = sin(3x°) repeats every 120°. Mixing those two conventions is the quietest way to get every answer wrong at once, which is why the calculator makes you pick an angle mode instead of guessing. If you’re rusty on how degree and radian inputs change trig values themselves, a sin cos tan calculator shows both side by side.
Reading a Graph Backwards: From Picture to Equation
Exams run this process in reverse: here’s a graph, write the equation. The derived formulas flip around neatly. Amplitude = (max − min)/2. Midline D = (max + min)/2 — the average of the extremes, not necessarily zero. Period = distance between two consecutive peaks. Phase shift = where the cycle starts, once you know the period.
Try it on a wave with maximum 5 at x = π/6 and minimum −1, next peak at x = 7π/6. Amplitude = (5 − (−1))/2 = 3. Midline D = (5 + (−1))/2 = 2. The peaks sit π apart, so the period is π and B = 2π/π = 2. A sine curve peaks a quarter period after its cycle starts, so the cycle begins at x = π/6 − π/4 = −π/12: a phase shift of π/12 to the left. Then C = B × shift = −π/6, and the equation assembles itself:
y = 3·sin(2x + π/6) + 2
Check it: 2x + π/6 = π/2 at x = π/6, so the peak lands where it should, at height 2 + 3 = 5. Punch the same coefficients into the calculator above — or sketch it on the graphing calculator — and the printed maximum, minimum, and shift confirm all four numbers.
A Full Cosine Example, Sign Traps Included
One worked run with every trap active at once: y = −2·cos(3x + π/4) − 1. Written against the standard form A·cos(Bx − C) + D, the coefficients are A = −2, B = 3, C = −π/4 (that plus sign again), D = −1. Line by line, exactly as the calculator prints it:
- Amplitude = |−2| = 2
- Period = 2π ÷ |3| = 2π/3 ≈ 2.094
- Phase shift = (−π/4) ÷ 3 = −π/12 → π/12 left
- Vertical shift = −1 → midline y = −1
- Max = −1 + 2 = 1, Min = −1 − 2 = −3
- A < 0 → reflected: starts at its minimum instead of its maximum
That last line deserves a sentence. An unreflected cosine starts each cycle at its peak. Negating A turns that opening peak into an opening trough, so this curve begins at (−π/12, −3) and climbs. Nothing about the amplitude changed — the wave is still 2 units tall from midline to extreme. The reflection and the amplitude are two separate facts extracted from the single symbol −2, and any trigonometry calculator worth using reports them separately.
Reference Table: Eight Sinusoids at a Glance
Worth a bookmark before a quiz — a spread of functions from bare sin(x) to fully loaded, with all four values worked out:
| Function | Amplitude | Period | Phase shift | Midline |
|---|---|---|---|---|
| y = sin(x) | 1 | 2π | none | y = 0 |
| y = 4·cos(x) | 4 | 2π | none | y = 0 |
| y = sin(2x) | 1 | π | none | y = 0 |
| y = 3·sin(2x − π/2) + 1 | 3 | π | π/4 right | y = 1 |
| y = cos(x − π/3) | 1 | 2π | π/3 right | y = 0 |
| y = −2·cos(3x + π/4) − 1 | 2 | 2π/3 | π/12 left | y = −1 |
| y = 5·sin(πx) | 5 | 2 | none | y = 0 |
| y = 0.5·sin(x/2 + π/8) − 3 | 0.5 | 4π | π/4 left | y = −3 |
The last row is the sneakiest: B = 1/2, so the small-looking π/8 becomes a phase shift of (−π/8)/(1/2) = −π/4. Dividing by a fraction enlarges the shift — a detail almost everyone meets for the first time as a lost exam point.
Where Students Lose Points
- Reporting C as the phase shift. For 3·sin(2x − π/2) + 1 that answer is π/2 instead of π/4 — off by a factor of B, every single time. Divide first.
- Flipping the direction. sin(2x + π/3) shifts left, not right. The standard form already contains a minus sign, so a visible + means C is negative.
- Dropping the absolute value on B. sin(−2x) has period π. Writing −π (or doubling to 4π after “fixing” the sign) turns one small slip into two.
- Calling the amplitude −2. Amplitude measures distance and can’t be negative. A = −2 means amplitude 2 plus a reflection across the midline — two facts, not one negative number.
- Mixing angle modes. Applying 2π/|B| to an equation whose x is in degrees gives a period of 6.28° instead of 180°. The formula must match the unit: 2π for radians, 360° for degrees.
Each of these fails loudly in the step-by-step box above, which is the honest reason to run your homework through the tool once: not to copy the answer, but to see which line of your own working diverges from the printed one.
