Lesson 1 of 3
Describe a repeating pattern with its midline, amplitude, period, and starting point.
Subject to changeOptional future study, outside the current course sequence. This lesson and its homework preview are unassigned and may change.
One idea at a time. Try the example, then keep going.
Big ideas · 1 of 8
Big ideas
Do this: Read each idea, then say it in your own words.
A sinusoidal model has amplitude and midline . Its range is [D-, D+].
For a nonconstant curve with A and B not equal to 0, the PeriodThe shortest positive input shift that repeats the whole pattern.Exactly: A fundamental period satisfies wherever defined, with no smaller positive period.Mix-up alert: For a nonconstant transformed sine or cosine, the period is ; for tangent it is .See this in Key words is in radians. The factored form shows the horizontal shift h directly.
Sketch a full cycle at quarter-period intervals. An unreflected cosine starts at a maximum; an unreflected sine starts on its midline, rising when .
A mathematical sine or cosine has all real inputs. A model may restrict time to or a stated observation window. State output units and do not assume predictions outside that window are reliable.
Step-through
Do this: Follow the labeled graph. Use the four steps to connect its points with the equation.
Model a small wheel whose center is 6 m high and radius is 2 m, starting at its highest point and turning once every 8 s.
Whole picture, 4 steps
4 steps. Tap a number or Show me how it works.
Worked example
Do this: Follow each step on paper before reading the result.
Worked example
Do this: Follow each step on paper before reading the result.
Mix-up alert
Do this: Explain the wrong idea and the repair.
Wrong idea: the phase shift of is . Repair: factor the input as ; the shift is right. Divide the whole constant by the input coefficient.
Tip
Do this: Use paper. State the definition before you calculate.
If a sinusoid ranges from 4 to 10, explain why its amplitude is 3 and its midline is 7.
Worked example
Do this: Work on paper and explain each step before checking it.
For :
Watch
Do this: Watch the public lesson, pause to try a step, then return to your practice.
The linked video is free to watch. The creator may offer optional paid resources; they are not needed for this lesson. If YouTube is blocked, use the lesson notes and worked examples here.
Ready when you are.
Activity 1 of 7 · 0 of 7 activities done
Flashcards
Do this: Say an answer before flipping. Use the hint when needed.
Card 1 of 8
0 Know it0 Still learning
Compare each extreme with the midline.
On the card: Enter or Space flips. Left and Right move. After revealing, 1 marks Still learning and 2 marks Know it. Tab reaches every control.
What is amplitude?
Half the distance between maximum and minimum output.
What is the midline for a range ?
.
What is the amplitude for a range ?
4.
What is the period of ?
radians.
What is the shift of ?
1 unit left.
What is the range of ?
.
How far apart are quarter-cycle sketch points?
One fourth of the period.
Does a physical model always allow every real time?
No. State its time interval and units.
True or false
Do this: Decide true or false, then read why.
Self-test
Do this: Work on paper, choose an answer, then read the feedback.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Self-test
Do this: Work on paper, choose an answer, then read the feedback.
6 questions. One at a time.
Practice, not a grade. No timer. Your best score stays in this browser.
Write it
Do this: Write a short explanation, then compare it with the model.
Your answer is not saved anywhere. It disappears when you leave the page.
Write it
Do this: Try the extension on paper, then explain your reasoning here.
Your answer is not saved anywhere. It disappears when you leave the page.
Match
Do this: Choose a word, then choose its meaning.
Connect the ideas
0 / 4 connectedChoose a term, then its meaning. Or drag its grip. Use Tab and Enter or Space; Escape clears.
Subject to changeOptional preview. This page does not mean this homework is assigned. Only turn it in if your teacher asks.
Optional practice preview · 3 points if assigned
Opening your homework…
Practice it: You did 0 of 7 activities.
You decide when you have finished. This is your study progress, not a grade.
Not marked finished yet.
