Learn with Amar
Teaching video
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Grade 11 chapters and video availability01 · Read and understand
What you will learn
- Read a factored sinusoidal model's amplitude, period, and phase shift.
- Justify the conclusion "The phase shift is π/6 to the right" using the stated assumptions.
Before you start
Sine graphs and transformations.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
Analyze y=2sin(3(x−π/6))+1.
Why this math matters
Read a factored sinusoidal model's amplitude, period, and phase shift. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- x is measured in radians for the trigonometric input.
- The formula describes the full real graph.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Separate horizontal shift from angular frequency
PausedQuestion: Start with the question. Paused.
Question
Start with the question
Analyze y=2sin(3(x−π/6))+1.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
Amplitude=2 and midline=1
The outside multiplier and addition control vertical behavior.
Work through the mathematics
The angular factor is three, so period=2π/3
A full input-phase change of 2π takes a shorter x interval.
Check the conclusion
The phase shift is π/6 to the right
Factoring the inside expression prevents confusing a phase angle with a horizontal shift.
The result
The phase shift is π/6 to the right
Factoring the inside expression prevents confusing a phase angle with a horizontal shift.
Common mistakes to catch
- The horizontal shift is not the unfactored constant inside sine.
- Amplitude is nonnegative and is the absolute outside multiplier.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What is y at x=π/6?
Show a hint
The sine input becomes zero.
Reveal answer and explanation
One
The graph crosses its midline there.
Practice 2
What is the range?
Show a hint
Add and subtract amplitude from the midline.
Reveal answer and explanation
[−1,3]
Sine ranges from −1 to one before scaling and shifting.
Take the idea with you
Label separately the phase angle and time delay in a periodic model.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Find area without first finding an altitude
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