0. Quick Reference — The Standard Tangent READ FIRST
GENERAL FORM:
\[ y = a\tan(bx - c) + d \]
| Letter | What it does | Memory word |
|---|---|---|
| a | Vertical stretch / flip | AMPLIFY |
| b | Changes period | BEATS (speed) |
| c | Horizontal (phase) shift | CRUISE (sideways) |
| d | Vertical shift | DRIFT (up/down) |
A · B · C · D
Amplify · Beats · Cruise · Drift
Say this aloud before every problem!
Say this aloud before every problem!
PERIOD = π/b
tan repeats every π (not 2π!)
New period = π ÷ b
New period = π ÷ b
ASYMPTOTES of y = tan(bx − c) occur where:
\[ bx - c = -\frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
👉 Solve for x to get each vertical asymptote location.
tan has NO amplitude — it goes to ±∞. The "a" just stretches how fast it climbs. Never say "amplitude = a" for tangent!
1. Period Changes y = tan(bx)
BEATS
Big b → squish graph (faster) · Small b → stretch graph (slower)
Period = π ÷ b
Period = π ÷ b
Worked Example A
Find the period of \( y = \tan(3x) \)
- Identify \(b\): here \(b = 3\)
- Period \(= \dfrac{\pi}{b} = \dfrac{\pi}{3}\)
- The graph repeats every \(\dfrac{\pi}{3}\) units.
Students often write Period = \(\dfrac{2\pi}{b}\) — that's for sin/cos, NOT tan! Tan period is \(\dfrac{\pi}{b}\).
1
What is the period of \( y = \tan(2x) \)?
period
★☆☆
2
What is the period of \( y = \tan\!\left(\dfrac{x}{2}\right) \)?
Careful! b is a fraction here. period ★☆☆
Careful! b is a fraction here. period ★☆☆
3
The period of \( y = \tan(bx) \) is \( \dfrac{\pi}{4} \). Find \(b\).
period
★★☆
2. Vertical Stretch & Reflection y = a·tan(x)
AMPLIFY
\(|a| > 1\) → steeper curve · \(0 < |a| < 1\) → flatter curve
Negative a → flips graph upside-down (REFLECT)
Negative a → flips graph upside-down (REFLECT)
Worked Example B
Describe how \(y = -3\tan(x)\) differs from \(y = \tan(x)\).
- \(a = -3\): negative → reflect over x-axis
- \(|a| = 3\) → vertically stretched by factor 3 (steeper)
- Period stays \(\pi\) (b = 1 unchanged)
4
Is the graph of \(y = -\tan(x)\) a reflection over the x-axis or y-axis?
reflect
★☆☆
5
Compare \(y = 5\tan(x)\) and \(y = \tan(x)\).
Which one rises faster near \(x = 0\)? stretch ★☆☆
Which one rises faster near \(x = 0\)? stretch ★☆☆
\(y = \tan(-x)\) is a reflection over the y-axis.
\(y = -\tan(x)\) is a reflection over the x-axis. These look similar but are DIFFERENT!
3. Vertical Shift y = tan(x) + d
DRIFT
\(+d\) → moves graph UP · \(-d\) → moves graph DOWN
Asymptotes do NOT move (they're still vertical!)
Asymptotes do NOT move (they're still vertical!)
Worked Example C
Describe \(y = \tan(x) - 2\).
- \(d = -2\) → entire graph shifts down 2 units
- The "middle" line (where the curve crosses) is now \(y = -2\)
- Asymptotes are still at \(x = \dfrac{\pi}{2} + n\pi\)
6
The graph of \(y = \tan(x) + 3\) has been shifted how many units, in which direction?
vertical shift
★☆☆
7
\(y = \tan(x) + k\) passes through the point \(\left(0,\, 5\right)\). Find \(k\).
vertical shift
★★☆
4. Phase (Horizontal) Shift y = tan(x − c)
CRUISE
\(y = \tan(x - c)\) → shifts graph RIGHT by c
\(y = \tan(x + c)\) → shifts graph LEFT by c
⚠️ The sign FLIPS! Minus = Right, Plus = Left
\(y = \tan(x + c)\) → shifts graph LEFT by c
⚠️ The sign FLIPS! Minus = Right, Plus = Left
TRICK: Set the inside = 0 and solve for x. That x is where the center of the graph moves.
e.g. \(\tan(x - \pi/4)\): set \(x - \pi/4 = 0\) → \(x = \pi/4\) → center moved right by \(\pi/4\)
e.g. \(\tan(x - \pi/4)\): set \(x - \pi/4 = 0\) → \(x = \pi/4\) → center moved right by \(\pi/4\)
Worked Example D
Find the phase shift of \(y = \tan\!\left(x + \dfrac{\pi}{3}\right)\)
- Inside is \(x + \dfrac{\pi}{3}\) — set equal to 0: \(x + \dfrac{\pi}{3} = 0\)
- Solve: \(x = -\dfrac{\pi}{3}\)
- Phase shift = \(\dfrac{\pi}{3}\) to the LEFT
8
What is the phase shift of \(y = \tan\!\left(x - \dfrac{\pi}{4}\right)\)?
phase shift
★☆☆
9
\(y = \tan(2x - \pi)\) — what is the phase shift?
Hint: factor out b first → \(\tan\!\left(2\left(x - \dfrac{\pi}{2}\right)\right)\) phase shift ★★☆
Hint: factor out b first → \(\tan\!\left(2\left(x - \dfrac{\pi}{2}\right)\right)\) phase shift ★★☆
5. Finding Asymptotes Vertical lines
BLOCKED
Asymptotes = where tan is BLOCKED (undefined)
Set: \(bx - c = \dfrac{\pi}{2} + n\pi\), solve for x
Set: \(bx - c = \dfrac{\pi}{2} + n\pi\), solve for x
Worked Example E
Find the first two positive asymptotes of \(y = \tan(2x)\)
- Set inside \(= \dfrac{\pi}{2} + n\pi\): so \(2x = \dfrac{\pi}{2} + n\pi\)
- Solve: \(x = \dfrac{\pi}{4} + \dfrac{n\pi}{2}\)
- \(n = 0\): \(x = \dfrac{\pi}{4}\) ✓ \(n = 1\): \(x = \dfrac{3\pi}{4}\) ✓
10
List the first two positive asymptotes of \(y = \tan(x)\).
asymptote
★☆☆
11
Find the asymptotes of \(y = \tan\!\left(x - \dfrac{\pi}{4}\right)\).
asymptote
phase shift
★★☆
Vertical shifts (d) do NOT affect asymptotes! Only b and c affect asymptote positions.
6. Combined Transformations All together!
STEP-BY-STEP ORDER (always do this!):
- Read off a, b, c, d from \(y = a\tan(bx - c) + d\)
- Period = \(\pi / b\)
- Phase shift = \(c / b\) (direction: same sign = right)
- Vertical shift = \(d\)
- Reflection? → if \(a < 0\)
Worked Example F — FULL ANALYSIS
Analyze \(y = -2\tan(3x - \pi) + 1\)
- \(a = -2\) → vertical stretch ×2, reflected over x-axis
- \(b = 3\) → Period \(= \dfrac{\pi}{3}\)
- \(c = \pi\) → Phase shift \(= \dfrac{\pi}{3}\) to the right
- \(d = 1\) → Graph shifts up 1
12
For \(y = 3\tan(2x) - 1\), state: (a) period, (b) vertical shift.
combo
★★☆
13
Fully describe the transformation of \(y = -\tan\!\left(\dfrac{x}{2} + \dfrac{\pi}{4}\right)\).
combo
reflect
★★☆
14
Which equation matches: period \(= 2\pi\), phase shift \(= \dfrac{\pi}{2}\) right, vertical shift down 3?
(A) \(y = \tan\!\left(\tfrac{x}{2} - \tfrac{\pi}{4}\right) - 3\) (B) \(y = \tan\!\left(\tfrac{x}{2} + \tfrac{\pi}{4}\right) - 3\) (C) \(y = \tan\!\left(2x - \pi\right) - 3\) combo ★★☆
(A) \(y = \tan\!\left(\tfrac{x}{2} - \tfrac{\pi}{4}\right) - 3\) (B) \(y = \tan\!\left(\tfrac{x}{2} + \tfrac{\pi}{4}\right) - 3\) (C) \(y = \tan\!\left(2x - \pi\right) - 3\) combo ★★☆
7. Sketching the Graph Draw it!
5-POINT METHOD
For one cycle of \(y = a\tan(bx)\), find 5 key x-values:
Left asymptote → \(-\frac{1}{4}\text{period}\) → 0 (center) → \(+\frac{1}{4}\text{period}\) → Right asymptote
y-values: ±∞ → \(-a\) → 0 → \(+a\) → ±∞
Left asymptote → \(-\frac{1}{4}\text{period}\) → 0 (center) → \(+\frac{1}{4}\text{period}\) → Right asymptote
y-values: ±∞ → \(-a\) → 0 → \(+a\) → ±∞
15
On the grid below, sketch ONE cycle of \(y = \tan(x)\) for \(-\dfrac{\pi}{2} < x < \dfrac{\pi}{2}\).
Label the asymptotes and the point at \(x = \dfrac{\pi}{4}\). sketch ★☆☆
Label the asymptotes and the point at \(x = \dfrac{\pi}{4}\). sketch ★☆☆
[ sketch your graph here ]
16
Sketch \(y = \tan(x) + 2\) and \(y = \tan(x)\) on the same axes.
What is the only difference between the two graphs? vertical shift ★★☆
What is the only difference between the two graphs? vertical shift ★★☆
[ sketch both graphs here ]
8. Tricky Problems — Watch Out! Most missed
🎯 These are the problems most students get wrong. Take your time — re-read each one twice before answering.
17
\(y = \tan(-2x)\) — is this the same as \(y = -\tan(2x)\)?
Hint: tan is an odd function: \(\tan(-\theta) = -\tan(\theta)\) tricky! ★★☆
Hint: tan is an odd function: \(\tan(-\theta) = -\tan(\theta)\) tricky! ★★☆
18
Two students analyze \(y = \tan(2x - \pi)\):
• Student A says phase shift \(= \pi\) (right)
• Student B says phase shift \(= \dfrac{\pi}{2}\) (right)
Who is correct, and why? phase shift tricky! ★★☆
• Student A says phase shift \(= \pi\) (right)
• Student B says phase shift \(= \dfrac{\pi}{2}\) (right)
Who is correct, and why? phase shift tricky! ★★☆
Phase shift formula is \(\dfrac{c}{b}\), NOT just \(c\). Always divide by \(b\)!
19
True or False: The graph of \(y = \tan(x) + 5\) has its asymptotes at
\(x = \dfrac{\pi}{2} + 5 + n\pi\). asymptote ★★☆
\(x = \dfrac{\pi}{2} + 5 + n\pi\). asymptote ★★☆
20
Write the equation of a tangent function with:
• period = \(\dfrac{\pi}{5}\) • shifted left \(\dfrac{\pi}{10}\) • flipped over x-axis • moved up 4 BOSS LEVEL ★★☆
• period = \(\dfrac{\pi}{5}\) • shifted left \(\dfrac{\pi}{10}\) • flipped over x-axis • moved up 4 BOSS LEVEL ★★☆
✨ Final Memory Cheat Sheet ABCD Summary
| Keyword | English Full Form | What changes | Example |
|---|---|---|---|
| AMPLIFY | Amplify / Reflect | Steepness, flip | \(y = 3\tan x\) → steep |
| BEATS | Beats = speed | Period = π/b | \(y = \tan(2x)\) → period π/2 |
| CRUISE | Cruise = slide sideways | Phase shift = c/b | \(y = \tan(x-\frac{\pi}{4})\) → right |
| DRIFT | Drift = float up/down | Vertical shift | \(y = \tan x + 2\) → up 2 |
| BLOCKED | Blocked = undefined | Asymptote location | \(bx-c = \frac{\pi}{2}+n\pi\) |
TAN ≠ SIN/COS
Tan period = π/b (not 2π/b!)
Tan has NO max/min (no amplitude)
Tan has vertical asymptotes
Tan has NO max/min (no amplitude)
Tan has vertical asymptotes
📝 My own notes:
✦ Good luck! You've got this. ✦