3.1.8 · D5Advanced Trigonometry
Question bank — Transformations of trig graphs — A·sin(Bx + C) + D (amplitude, period, phase, vertical shift)
True or false — justify
Doubling doubles the period.
False. The period is , so doubling halves the period — a faster input runs through one cycle in less .
A negative gives a wave with negative height.
False. Amplitude is , always ; the minus sign flips the wave across the midline (a reflection), it does not shrink or "negate" the height.
The midline of is always .
False. The midline is ; it is only when . The wave oscillates around , between and .
and are the same graph.
True. Shifting left by lands exactly on — this is cos as shifted sin, and both have amplitude 1, period , midline 0.
For , the maximum value is always .
==False (unless )==. The maximum is . If then , but the peak still reaches ; use the absolute value.
Changing changes how tall the wave is.
False. (and ) live inside the sine and only affect the horizontal picture — the slide. Height is set by , which is outside.
and are identical graphs.
True. Adding to the input shifts by a whole period ( one full period), which lands the wave back on itself.
If two sinusoids have the same but different , they can never coincide.
False. If the two values differ by a multiple of (i.e. shifts differing by whole periods), the graphs coincide exactly.
Increasing makes the wave steeper.
False. only lifts the whole graph vertically; steepness (rate of climb) depends on and , not on the midline height.
Spot the error
", so the phase shift is (left by )."
Error: forgot to divide by . Factor first: , so the shift is (right), not .
"Since is inside, shifts right by 1."
Error: inside changes act opposite to intuition. moves the graph left by 1; a rightward slide needs .
" has amplitude ."
Error: amplitude is a distance, so it is . The minus is a reflection across the midline, not a negative height.
"For , the period is ."
Error: belongs in the denominator: . Larger means shorter period, not longer.
" reaches a minimum of ... wait, of , but also touches ."
Error: the wave never touches . Min ; the whole wave sits above the -axis.
"To graph , apply the vertical shift first, then squeeze horizontally."
Error of order: inside operations (, ) act on and should be handled together (horizontal), then outside (, ) act on (vertical). Mixing the order can misplace the phase shift.
" starts (at ) at the midline going up, just like ."
Error: starts at a maximum at , not the midline. It is shifted left by a quarter period, so its starting behaviour differs.
Why questions
Why is (not ) used in the period formula?
A negative still gives a real, positive repeat-distance; reflects the wave but leaves the cycle length unchanged, so we take .
Why must we factor out of before reading the phase shift?
The slide is measured in units of , but is added to . Only exposes the true horizontal shift ; reading directly ignores the horizontal squish.
Why does a bigger produce a "tighter," more crowded wave?
The input reaches over a smaller stretch of , so one full cycle is completed sooner and the peaks pack closer together.
Why is amplitude defined as a distance from the midline rather than from zero?
When the wave oscillates around , not ; the meaningful "height" is how far it swings from its own centre, which is regardless of .
Why can any cosine wave be rewritten as a sine wave (and vice versa)?
They are the same shape offset by a quarter period; a horizontal shift of in the input converts one into the other, so a phase term absorbs the difference (see cos as shifted sin).
Why do we need exactly four parameters and not three or five?
They are the four independent freedoms of a sinusoid — height, repeat-rate, horizontal start, and vertical level. Fewer cannot describe a general wave; a fifth would duplicate one of these (see Function transformations).
Why does swapping max↔min still leave the period and midline unchanged?
A reflection across the midline flips which extreme comes first but doesn't stretch time (period) or move the centre line (); only the vertical orientation flips.
Edge cases
What happens to when ?
The wave collapses to the constant line — a flat "degenerate" sinusoid with zero amplitude, no oscillation, no meaningful period.
What is the graph when ?
The input becomes the constant , so is a horizontal line; with no inside, there is no cycle and the period is undefined (division by zero).
If exactly, what is special about the graph?
The minimum just touches the -axis () at the bottom of each trough; the wave is tangent to the axis there and never goes negative.
For a purely negative amplitude like , where is the graph at ?
At , so (the midline), but it heads downward first instead of upward — the reflection reverses the starting direction, not the starting value.
What is the phase shift of (no term)?
Zero — with , , so the wave starts exactly like the basic : at the midline rising through it (as on the Unit circle and sine definition).
How does the graph behave as (very large )?
The period , so the wave oscillates infinitely fast — the peaks bunch together and become indistinguishable; a limiting, non-physical "infinitely crowded" wave.
For very small near zero, what does one period look like?
The period becomes enormous, so over any ordinary window the graph looks like a slow, nearly flat drift — you barely see one cycle. This links to slow oscillations in Simple harmonic motion.
Connections
- Unit circle and sine definition — the source of 's range and starting behaviour.
- Function transformations — the general rules these traps specialise.
- cos as shifted sin — the quarter-period identity behind several true/false items.
- Simple harmonic motion — where slow/fast oscillation limits appear physically.
- Solving trig equations — the natural next step once you can read off .