Single-wave features
Factor the argument before reading the shift
- Amplitude The magnitude of the outer coefficient
- Period 2π/n for positive frequency n
- Range [k−R,k+R] over a full cycle or all real x
- Shift R sin(nx+φ)=R sin(n(x+φ/n)), the signed horizontal shift is −φ/n
Worked example
Read the features of y=4 cos(3x−π/2)+2
\[y=4\cos(3(x-\pi/6))+2\]
- \(\displaystyle R=4,\quad T=\frac{2\pi}{3},\quad h=\frac\pi6,\quad -2\le y\le6\)The phase constant π/2 is divided by 3 to obtain the horizontal shift
Common mistakes
Wrong shift
✕nx+φ means a left shift φ✓factor n: the signed horizontal shift is −φ/n
The horizontal coordinate is multiplied by nDifficulty levels
introRead exact parameters and features
Questions about this topic
Does an arbitrary restricted interval contain both extrema?+
No. Check whether the interval includes the relevant phase positions
Can I use a calculator?+
No. This topic is practised without a calculator
Which year level and chapter is this?+
Year 11, VCE Units 1 & 2, Specialist Mathematics, chapter 7: Trigonometry and transformations
In this chapter
Convert angle measure and use the radian arc relationArea of a sectorUse the sine rule in a non-right-angled triangleUse the cosine rule in a non-right-angled triangleTrigonometry in three dimensionsExact circular-segment area and perimeterChoose perpendicular sections and calculate spatial anglesUse exact trigonometric identities and quadrant informationCompound, double-angle and reciprocal identitiesTransform and reconstruct sine and cosine functionsCombine waves and convert trigonometric sums and productsMap or recover points and finite coordinate setsCompose two plane transformations in orderRotate, reflect and interpret fixed points and area