GRAPHING TRIG FUNCTIONS MOST IMPORTANT THING

SYMMETRY 

Cos(-x) = Cos (x)  even. symetrical about y-axis Graphically: mirror image across the y-axis.

-SIn(x) = Sin(-x) odd. symetrical about origin. Graphically: rotate the graph 180° about the origin, and it looks the same.

Function	Symmetry	Graph Feature
sin x	Odd	Origin symmetry; wave crosses origin
cos x	Even	y-axis symmetry; wave starts at max
tan x	Odd	Origin symmetry; vertical asymptotes every π
cot x	Odd	Origin symmetry; vertical asymptotes every π
sec x	Even	y-axis symmetry; vertical asymptotes where cos x = 0
csc x	Odd	Origin symmetry; vertical asymptotes where sin x = 0

y=Sinx

🌀 1⃣ Sine Function: y = a sin(bx + c) + d

Smooth wave starting at the origin (goes up first).

ODD FUNCTION

Amplitude: 1

Period: 2π

Range: [−1,1]

Range: [−infinity,infinity ]

Y-intercept: (0, 0)

Max: 1 at x=π/2

Min: −1 at x=3π/2

Key points 0 π/2, π, 3π/2, 2π

(0,0),(π/2​,1),(π,0),(3π​/2,−1),(2π,0)

Practice problem: F(x)= 3sin(2x-π/4)

transformations

Property	Formula	Meaning
Amplitude	just number in front will always be positive	a
Period	2π/b	b
Phase shift	-c/b or set inside equal to 0 and solve	Move left/right
Vertical shift	(d) at end	Move up/down

Reflection over x-axis: y= - sin(x)

Reflection over y-axis: y=sin⁡(−x)

y=cosx

2⃣ Cosine Function: y=acos⁡(bx+c)+d

EVEN FUNCTION

Starts at a maximum instead of 0.

• Amplitude: 1

• Period: 2π

• Range: [−1,1]

• Y-intercept: (0, 1)

• Max: 1 at x=0 

• Min: -1 at x=π

transformations are the same as SIN

Key points

Max → Mid → Min → Mid → Max

(0,1),(π/2,0),(π,−1),(3π/2,0),(2π,1)

Practice problem 

F(x)=-3cos(x-π)+1

y= tan(x)

🔺 3⃣ Tangent Function: y = a tan(bx + c) + d Base function

ODD SYMMETRY

• Period: π

• Amplitude: none (goes to infinity)

• Y-intercept: (0, 0)

• Vertical asymptotes: at x=π/2+nπ

• Range: (−∞,∞)

• No horizontal asymptote

  Property	Formula	Meaning
  Period	π/b	b
  Phase shift	-c/b	Left/right
  Vertical shift	(d)	Move up/down

Key points (1 period): (−π/2,VA),(0,0),(π/2,VA)

y=cot(x)

⚡ 4⃣ Cotangent Function: y = a cot(bx + c) + d

ODD SYMMETRY

• Period: π

• Vertical asymptotes: at x=nπ

• No amplitude

• Range: (−∞,∞)

Transformations:

Same pattern as tangent:

• Period: π/b

• Phase shift: −c/b

• Vertical shift: d

Key points:

Crosses through midline halfway between asymptotes, but decreases instead of increases.

y=cscx

🌊 5⃣ Cosecant Function: y = a csc(bx + c) + d

This is the reciprocal of sine, Smart to graph sin of the equation firsty to see.

it is undefined where sin x = 0.

• Vertical asymptotes: at x=nπ

• Period: 2π

• No amplitude

• Range: (−∞,−1]∪[1,∞)

• Y-intercept: undefined (since sin 0 = 0)

Shape:

Curves that mirror 1/sin⁡x: open “U” and “n” shapes between asymptotes.

y=sec(x)

🌤 6⃣ Secant Function: y = a sec(bx + c) + d

Reciprocal of cosine, so undefined where cos x = 0.

Smart to graph cos(x) first

Vertical asymptotes: at x=π/2+nπ

• Period: 2π

• No amplitude

• Range: (−∞,−1]∪[1,∞)

• Y-intercept: (0, 1)

Shape:

Curves that mirror 1/cos⁡x: alternating U and n shapes.

Tricks to graph #1

1⃣ Use Symmetry to Your Advantage

• Even functions (cos, sec) → symmetric across y-axis → you only need to graph the right half.

• Odd functions (sin, tan, cot, csc) → symmetric around the origin → you can graph one side and rotate 180° for the other side.

Tricks to graph #2

2⃣ Memorize Key Angles

• Unit circle angles: 0°, 30°, 45°, 60°, 90° (or 0, π/6, π/4, π/3, π/2)

• Sine & cosine values:

0→0,π/6→1/2,π/4→2/2,π/3→3/2,π/2→10 \to 0, \quad \pi/6 \to 1/2, \quad \pi/4 \to \sqrt{2}/2, \quad \pi/3 \to \sqrt{3}/2, \quad \pi/2 \to 10→0,π/6→1/2,π/4→2​/2,π/3→3​/2,π/2→1

• Tangent values: ratio of sine/cosine → easy to remember 0, 1/√3, 1, √3, undefined

Tricks to graph #3

3⃣ Use Reciprocal Relationships

• sec = 1/cos, csc = 1/sin, cot = 1/tan

• When graphing sec or csc, start with cos or sin, then invert the curve to get sec/csc branches.

Tricks to graph #4 really good one

4⃣ Period Shortcuts

• Sine/cosine period = 2π/b

• Tangent/cotangent period = π/b

• Sec/csc follow the same period as their base function (cos/sin)

• Tip: divide the period into 4 equal sections to plot key points quickly.

Tricks to graph #5

5⃣ Vertical & Horizontal Shifts

• Graphs are always relative to the midline (y = d)

• Max = midline + amplitude

• Min = midline - amplitude

Tricks to graph #6

7⃣ Asymptotes

• Tangent/cotangent: vertical asymptotes every period interval (where cos or sin = 0)

• Sec/csc: vertical asymptotes where their base function = 0

Last trick to graph

Bonus Tip

• When stuck, always start with base function: sin, cos, tan

• Apply amplitude, period, phase shift, vertical shift last

• Add reflections and asymptotes after plotting key points

Test problem number 1'

f(x)=3sin(2x-π/4)

do y intercept as well

Test problem number 2

f(x)= sec(x/2)+2

do y-intercept as well