Key-point sketches and transformed tangent domains
Введено| English | Значение |
|---|---|
| asymptote/ˈæsɪmptəʊt/ | асимптота |
Which input angles locate its peaks, crossings and breaks?
- A shifted and stretched trig graph can hide its key points.
- This lesson studies asymptote 渐近线: A line approached by a graph while its values grow without bound or tend toward a limit.
The method
- For a sine/cosine graph a f(kx+c)+d, amplitude is |a|, centre line y=d and period 2π/|k| for k≠0.
- Transform the five quarter-cycle base inputs to locate key points.
- A negative a reflects vertically.
- For tangent the period is π/|k|, there is no amplitude and the range is all real numbers; exclude every input giving a base-angle π/2+jπ.
Why does y=tan(2x+π/3) have no amplitude and no joinable branches?
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
Tangent has no maximum or minimum - "amplitude" is meaningless for it. Each branch rises between asymptotes x=π/12+jπ/2, and joining across an asymptote invents points that do not exist.
A worked case
For y=2cos(x−π/3)−1, one cycle has points (π/3,1),(5π/6,−1),(4π/3,−3),(11π/6,−1),(7π/3,1). The range is [−3,1] and period 2π. For y=−3sin2x+1, the points x=0,π/4,π/2,3π/4,π give y=1,−2,1,4,1; the period is π and range [−2,4]. For y=tan(2x+π/3), zeros satisfy 2x+π/3=jπ, hence x=jπ/2−π/6. Asymptotes satisfy 2x+π/3=π/2+jπ, hence x=π/12+jπ/2. On 0≤x<π, the zeros are π/3 and 5π/6 and the asymptotes π/12 and 7π/12.
Key-point sketches and transformed tangent domains
For a sine/cosine graph a f(kx+c)+d, amplitude is |a|, centre line y=d and period 2π/|k| for k≠0
Justify each triangle candidate or transformed key point against its defining conditions.
Find the maximum of 2cos(x−π/3)−1.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Cosine can attain 1, giving 2×1−1=1.
Where it goes wrong
- The cosine maximum need not occur at x=0 after a horizontal shift.
- For the stated tangent function, draw separate increasing branches between asymptotes.
- A negative horizontal or vertical factor reverses the direction; never join branches across a break.
- Vertical translations change outputs but not the period or excluded x values.
- A displayed interval may contain only part of a complete shifted cycle.
Not true: A transformed tangent graph has a finite amplitude and can be joined across its vertical asymptotes. A tangent graph has no amplitude, and its branches must not be joined across asymptotes.
Find its minimum.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Cosine can attain −1, giving 2×(−1)−1=−3.
A transformed tangent graph has a finite amplitude and can be joined across its vertical asymptotes.
Сложность 1/5 · 100 монет + небольшой бонус за первое выполнение сегодня
A tangent graph has no amplitude, and its branches must not be joined across asymptotes.
Applying it
- Find period and centre line first, then map exact base-angle key points.
- Add enough repeats to cover the requested interval and label scale, extremes and asymptotes.
- To find x-axis crossings, solve the output equation rather than assuming every centre-line crossing has y=0.
- Check a sample point in each tangent branch and preserve the original domain.
Write the period of tan(2x+π/3) as kπ. Find k.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Tangent has base period π; dividing by input coefficient 2 gives π/2, hence k=1/2.
Match each transformation feature of a f(kx+c)+d to its effect on a sine graph.
Сложность 2/5 · 140 монет + небольшой бонус за первое выполнение сегодня
Transform the five quarter-cycle key points; a negative a reflects vertically, and shifts move the maximum away from x=0.
In your course
- AQA A-level Mathematics (7357), specification section E.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Asymptote: A line approached by a graph while its values grow without bound or tend toward a limit.
- For a sine/cosine graph a f(kx+c)+d, amplitude is |a|, centre line y=d and period 2π/|k| for k≠0.
- Watch for: the cosine maximum need not occur at x=0 after a horizontal shift.