Amplitude, Period and Phase Shift
y = A sin(B(x − C)) + D transforms the wave.
About this topic
For y = A sin(B(x − C)) + D: amplitude is the absolute value of A, the height from midline to peak; period is 2π divided by the absolute value of B, the length of one cycle; phase shift C moves the wave horizontally; and D shifts the midline vertically. Negative A reflects the wave.
Reading these four parameters lets you sketch any sinusoid quickly: draw the midline y = D, mark the maximum and minimum at D plus or minus the amplitude, then divide one period into quarters starting at x = C to place the key points. Factoring out B first is essential, since the shift is C and not the number left inside the bracket.
Example: y = 3 sin(2(x − π/4)) + 1 has amplitude 3, period π, shift of π/4 to the right and a midline at y = 1, so it ranges from −2 to 4. A tide that rises 2 m above and below mean sea level with a 12-hour cycle has amplitude 2 and B = 2π/12 = π/6.
These parameters describe sound loudness and pitch, daylight hours over a year and AC voltage. A common error is saying the period is B instead of 2π/B, or reading the shift as 2 in a bracket like (2x − π/2) without factoring out the 2 first.
Key ideas
- A sets height
- Period = 2π/B
- C shifts horizontally and D vertically
Quick check
Use cases: where Amplitude, Period and Phase Shift helps
- Homework: review the key idea behind Amplitude, Period and Phase Shift before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Amplitude, Period and Phase Shift without notes.
- Visual learning: play with the interactive lab to see how Amplitude, Period and Phase Shift changes when you adjust its inputs.
- Teaching a friend: explain Amplitude, Period and Phase Shift in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Amplitude, Period and Phase Shift?
y = A sin(B(x − C)) + D transforms the wave.
What will I learn in Amplitude, Period and Phase Shift?
A sets height. Period = 2π/B. C shifts horizontally and D vertically.
Is Amplitude, Period and Phase Shift part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Amplitude, Period and Phase Shift lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Amplitude, Period and Phase Shift?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.