Unit 6 Worksheet 18 Graphing Sine Functions | PDF - Free Printable

Let’s solve each problem step by step.

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First, recall the standard form:

> y = a sin(bx - c) + d

From this, we can find:

- Amplitude = |a| → how tall the wave is from center to peak
- Period = 2π / |b| → how long one full cycle takes
- Phase Shift = c / b → how far left or right the graph shifts (positive = right, negative = left)
- Initial Interval = [phase shift, phase shift + period] → where one full cycle starts and ends
- Vertical Shift = d → moves the whole graph up or down (not asked here, but good to know)

Also, for the parent function y = sin x:
- Amplitude = 1
- Period = 2π
- Phase Shift = 0
- Initial Interval = [0, 2π]

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Practice Problem 1:

y = 2 sin(x - π/4)

Compare to standard form:
y = a sin(bx - c) + d
→ Here, a = 2, b = 1, c = π/4, d = 0

Now calculate:

- Amplitude = |a| = |2| = 2
- Period = 2π / |b| = 2π / 1 = 2π
- Phase Shift = c / b = (π/4) / 1 = π/4 → to the RIGHT
- Initial Interval = [phase shift, phase shift + period] = [π/4, π/4 + 2π] = [π/4, 9π/4]

*(Note: π/4 + 2π = π/4 + 8π/4 = 9π/4)*

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Practice Problem 2:

y = sin(x + π)

Rewrite to match standard form:
y = sin(x + π) = sin(1·x - (-π)) → so c = -π

So: a = 1, b = 1, c = -π, d = 0

Calculate:

- Amplitude = |1| = 1
- Period = 2π / 1 = 2π
- Phase Shift = c / b = (-π) / 1 = -π → which means shift LEFT by π
- Initial Interval = [phase shift, phase shift + period] = [-π, -π + 2π] = [-π, π]

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✔ All calculations checked.

Final Answer:

For y = 2 sin(x - π/4):
- Amplitude: 2
- Period: 2π
- Phase Shift: π/4
- Initial Interval: [π/4, 9π/4]

For y = sin(x + π):
- Amplitude: 1
- Period: 2π
- Phase Shift: -π
- Initial Interval: [-π, π]