Trigonometric graphs repeat in cycles. The main parent graphs are sine, cosine, and tangent.
Their transformed form is usually written as:
\[
y=A\,f(Bx-C)+D
\]
where \(f(x)\) is \(\sin x\), \(\cos x\), or \(\tan x\).
1. Amplitude
For sine and cosine, amplitude is the distance from the midline to the maximum or minimum.
\[
\text{Amplitude}=|A|
\]
Tangent does not have amplitude because it is unbounded.
2. Period
The period is the horizontal length of one complete cycle.
For sine and cosine:
\[
T=\frac{2\pi}{|B|}
\]
For tangent:
\[
T=\frac{\pi}{|B|}
\]
3. Phase shift
The phase shift tells how far the graph moves left or right.
\[
h=\frac{C}{B}
\]
If \(h\) is positive, the graph shifts right. If \(h\) is negative, it shifts left.
4. Vertical shift and midline
The value \(D\) moves the graph up or down.
\[
\text{midline: } y=D
\]
For sine and cosine:
\[
y_{\max}=D+|A|
\]
\[
y_{\min}=D-|A|
\]
5. Tangent asymptotes
Tangent has vertical asymptotes where the tangent expression is undefined.
For:
\[
y=A\tan(Bx-C)+D
\]
the asymptotes satisfy:
\[
Bx-C=\frac{\pi}{2}+k\pi
\]
\[
x=\frac{C+\pi/2+k\pi}{B}
\]
6. Example
For:
\[
y=3\sin(2x-\pi/3)+1
\]
we have \(A=3\), \(B=2\), \(C=\pi/3\), and \(D=1\).
\[
\text{Amplitude}=|3|=3
\]
\[
T=\frac{2\pi}{2}=\pi
\]
\[
h=\frac{C}{B}=\frac{\pi/3}{2}=\frac{\pi}{6}
\]
\[
\text{midline: }y=1
\]
\[
y_{\max}=1+3=4,\qquad y_{\min}=1-3=-2
\]
7. Formula summary
| Feature |
Sine / cosine |
Tangent |
| General form |
\(y=A\sin(Bx-C)+D\) or \(y=A\cos(Bx-C)+D\) |
\(y=A\tan(Bx-C)+D\) |
| Amplitude |
\(|A|\) |
Not defined |
| Period |
\(\dfrac{2\pi}{|B|}\) |
\(\dfrac{\pi}{|B|}\) |
| Phase shift |
\(\dfrac{C}{B}\) |
\(\dfrac{C}{B}\) |
| Vertical shift |
\(D\) |
\(D\) |
| Midline |
\(y=D\) |
\(y=D\) |
| Asymptotes |
None |
\(Bx-C=\dfrac{\pi}{2}+k\pi\) |
8. Common mistakes
- Do not confuse \(C\) with the phase shift. The phase shift is \(C/B\), not just \(C\).
- Use \(2\pi/|B|\) for sine and cosine, but \(\pi/|B|\) for tangent.
- Tangent graphs do not have maximum, minimum, or amplitude.
- A negative \(A\) reflects the graph across its midline.
- The vertical shift \(D\) gives the midline \(y=D\), not the amplitude.
