Graphing Trig Functions - Kuta Software - Free Printable

Let’s solve each problem step by step. We’ll graph cosine functions by understanding how the equation changes the basic cosine wave:
Basic cos(x) starts at (0,1), goes down to (π, -1), back up to (2π, 1).
We adjust for:
- Horizontal shift: inside the parentheses with x → moves left/right
- Vertical shift: outside → moves up/down
- Amplitude & reflection: coefficient in front → stretches/flips

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Problem 1: y = cos(x + π/4) – 2

Step 1: Start with basic cos(x) — max at x=0, min at x=π, etc.

Step 2: “x + π/4” means shift LEFT by π/4. So instead of starting at x=0, it starts at x = -π/4.

Step 3: “–2” means shift DOWN 2 units. So the whole graph moves down.

Key points after shifting:

Original key points of cos(x):
(0, 1), (π/2, 0), (π, -1), (3π/2, 0), (2π, 1)

Shift left by π/4 → subtract π/4 from x-values:
(-π/4, 1), (π/4, 0), (3π/4, -1), (5π/4, 0), (7π/4, 1)

Then shift down 2 → subtract 2 from y-values:
(-π/4, -1), (π/4, -2), (3π/4, -3), (5π/4, -2), (7π/4, -1)

Plot these and connect smoothly.

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Problem 2: y = cos(x + π) + 5

Step 1: “x + π” → shift LEFT by π.

Step 2: “+5” → shift UP 5.

Original key points:
(0,1), (π/2,0), (π,-1), (3π/2,0), (2π,1)

Shift left by π:
(-π,1), (-π/2,0), (0,-1), (π/2,0), (π,1)

Shift up 5:
(-π,6), (-π/2,5), (0,4), (π/2,5), (π,6)

Note: cos(x + π) is same as -cos(x), so this is a flipped cosine shifted up 5.

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Problem 3: y = cos(x – π/2) + 3

Step 1: “x – π/2” → shift RIGHT by π/2.

Step 2: “+3” → shift UP 3.

Original points:
(0,1), (π/2,0), (π,-1), (3π/2,0), (2π,1)

Shift right by π/2:
(π/2,1), (π,0), (3π/2,-1), (2π,0), (5π/2,1)

Shift up 3:
(π/2,4), (π,3), (3π/2,2), (2π,3), (5π/2,4)

Also note: cos(x – π/2) = sin(x), so this is just sine wave shifted up 3.

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Problem 4: y = cos(x – π) + 4

Step 1: “x – π” → shift RIGHT by π.

Step 2: “+4” → shift UP 4.

Original points:
(0,1), (π/2,0), (π,-1), (3π/2,0), (2π,1)

Shift right by π:
(π,1), (3π/2,0), (2π,-1), (5π/2,0), (3π,1)

Shift up 4:
(π,5), (3π/2,4), (2π,3), (5π/2,4), (3π,5)

Also: cos(x – π) = -cos(x), so flipped cosine shifted up 4.

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Problem 5: y = –3 · cos(–²⁄₃ x)

First, simplify: cos is even function → cos(–θ) = cos(θ), so:

y = –3 · cos(²⁄₃ x)

Step 1: Amplitude = |–3| = 3 → graph goes from y = –3 to y = 3? Wait — no! The negative sign flips it vertically.

Actually: since it’s –3 * cos(...), the max becomes –3 and min becomes 3? Let’s think:

Normal cos ranges from –1 to 1.

Multiply by –3: ranges from –3*(1) = –3 to –3*(-1) = 3 → but wait, that’s backwards.

Actually: when you multiply by negative, it flips upside down.

So if original cos has max 1 → now becomes –3*1 = –3 (minimum)

Original min –1 → becomes –3*(-1) = 3 (maximum)

So amplitude is 3, but inverted.

Step 2: Period change: inside is (²⁄₃)x → period = 2π / (²⁄₃) = 2π * ³⁄₂ = 3π

So one full cycle every 3π.

Key points for cos(²⁄₃ x):

At x=0: cos(0)=1 → y = –3*1 = –3

At x = (3π)/4: (²⁄₃)(3π/4) = π/2 → cos(π/2)=0 → y=0

At x = (3π)/2: (²⁄₃)(3π/2) = π → cos(π)=–1 → y = –3*(–1) = 3

At x = (9π)/4: (²⁄₃)(9π/4) = 3π/2 → cos=0 → y=0

At x = 3π: (²⁄₃)(3π) = 2π → cos=1 → y=–3

So key points:
(0, –3), (3π/4, 0), (3π/2, 3), (9π/4, 0), (3π, –3)

Plot these.

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Problem 6: y = –2 · cos(²⁄₃ x)

Similar to #5, but amplitude 2, not 3.

Period still 3π.

cos(²⁄₃ x) at:

x=0 → cos(0)=1 → y = –2*1 = –2

x=3π/4 → cos(π/2)=0 → y=0

x=3π/2 → cos(π)=–1 → y = –2*(–1) = 2

x=9π/4 → cos(3π/2)=0 → y=0

x=3π → cos(2π)=1 → y=–2

Key points:
(0, –2), (3π/4, 0), (3π/2, 2), (9π/4, 0), (3π, –2)

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Final Answer:

Since we are asked to graph, and cannot draw here, I will describe the key points for each graph that should be plotted and connected smoothly.

Problem 1: Plot (-π/4, -1), (π/4, -2), (3π/4, -3), (5π/4, -2), (7π/4, -1)

Problem 2: Plot (-π, 6), (-π/2, 5), (0, 4), (π/2, 5), (π, 6)

Problem 3: Plot (π/2, 4), (π, 3), (3π/2, 2), (2π, 3), (5π/2, 4)

Problem 4: Plot (π, 5), (3π/2, 4), (2π, 3), (5π/2, 4), (3π, 5)

Problem 5: Plot (0, -3), (3π/4, 0), (3π/2, 3), (9π/4, 0), (3π, -3)

Problem 6: Plot (0, -2), (3π/4, 0), (3π/2, 2), (9π/4, 0), (3π, -2)

Connect each set of points with a smooth cosine-shaped curve, respecting the period and direction (flipped if negative amplitude).

✔ All graphs are based on transforming the standard cosine wave using shifts, reflections, and scaling.