g10-07-11 · Core topics
Phase is divided by frequency
Does adding φ shift every sine graph by−φ?
1 MIN · PREDICT ON PAPER
Compare sin(x+90°) andsin(2x+90°), using x in degrees for this comparison.
Discuss or write first. No typed answer is required to continue.
The shift is −φ/ω, not just −φ.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
Factoring ωx+φ=ω(x+φ/ω) shows horizontal shift−φ/ω. Phase and horizontal displacement are different quantities.
Conditions: ω≠0 is required for a horizontal-shift interpretation; convert φ to radians before plotting against radian x.
Mathematical justification: Use algebraic factorisation inside the sine, then apply the definition of horizontal translation to x−h.
1 MIN · EXIT CHECK
Find the shift ofsin(2x+π/2).
Check after answering
−π/4, meaning π/4 to the left.
Teacher guidance and extension
Before: Angles, coordinates and basic function graphs.
Ask: Where is the input at which the inner angle becomes0?
Watch for: Shifting by−φ even when the frequency is not1.
Challenge: Predict the shift if ω is negative.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
Optional local notebook
Paper responses are enough. Drafts remain in this tab; choose local saving to include them in the existing complete backup.
Complete backup and restore