Transformations and Periodic Models

Develop phase shift in factored form, choosing a midline and model.

Lesson 2 of 42 parts Practice Flashcards

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Phase Shift in Factored Form

Objective: Explain phase shift in factored form, apply the rule, and justify relevant restrictions.

In y=A sin(B(x−h))+D with B≠0, h is the horizontal shift and D is the midline. For an unfactored argument Bx−C, factor out B to obtain h=C/B. Reading C directly as the shift confuses angle units with input units. Choose five quarter-cycle reference points to sketch a full cycle, then extend the pattern.

Formulas

$A sin(Bx−C)+D=A sin(B(x−C/B))+D$

Worked examples

Try itFor y=3 sin(5(x−7))+1, find the horizontal shift in radians (right positive).Show answer

Step 1: Use factored form A sin(B(x−h))+D. Step 2: The value of h is 7. Step 3: The shift is 7 radians to the right, not 35.

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Phase Shift in Factored Form

Objective: Explain phase shift in factored form, apply the rule, and justify relevant restrictions.

Formulas

$A sin(Bx−C)+D=A sin(B(x−C/B))+D$

Worked examples

Try itFor y=3 sin(5(x−7))+1, find the horizontal shift in radians (right positive).Show answer

Step 1: Use factored form A sin(B(x−h))+D. Step 2: The value of h is 7. Step 3: The shift is 7 radians to the right, not 35.

Choosing a Midline and Model

Objective: Explain choosing a midline and model, apply the rule, and justify relevant restrictions.

For a sinusoidal model with observed extrema M and m, choose midline D=(M+m)/2 and amplitude |A|=(M−m)/2. A known repeat interval gives B=2π/T; a reference maximum or minimum helps determine phase. These features do not justify assuming perfect periodicity outside the observation window. Check units, contextual range, and residual differences before extending a prediction.

Formulas

$D=(M+m)/2; |A|=(M−m)/2$

Worked examples

Try itA sinusoidal model has maximum 19 and minimum 5. Find its midline value.Show answer

Step 1: Average maximum and minimum. Step 2: Compute (19+5)/2. Step 3: The midline is y=12.

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