Sine Cosine Graphs

Grade 12 · trigonometry · 100 practice problems · read aloud

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📈 Understanding Sine and Cosine Graphs

Sine and Cosine functions are periodic functions that model repetitive phenomena like sound waves, light waves, and seasonal patterns. Understanding their graphs is crucial for analyzing amplitude, frequency, and phase shift in real-world applications.

🔍 Key Features & Parameters

The standard form is y = A sin(B(x - C)) + D or y = A cos(B(x - C)) + D.

  • |A| = Amplitude (vertical stretch/compression)
  • B affects the Period: Period = 2π / |B|
  • C = Phase Shift (horizontal translation)
  • D = Vertical Shift

📝 Step-by-Step Graphing Guide

  1. Identify the Amplitude (|A|): This is the height from the centerline to the peak.
  2. Calculate the Period: Use the formula Period = 2π / |B|.
  3. Determine the Phase Shift: The graph shifts right if C > 0, left if C < 0.
  4. Find the Vertical Shift (D): This moves the entire graph up or down.
  5. Plot Key Points: For sine, start at the midline; for cosine, start at the maximum (if A>0). Divide the period into 4 equal parts to mark one complete cycle.

Example 1: Graph y = 3 sin(2x)

  1. Amplitude: |A| = |3| = 3
  2. Period: 2π / |2| = π
  3. Phase Shift: None (C=0)
  4. Vertical Shift: None (D=0)
  5. Key points for one period (0 to π): (0,0), (π/4, 3), (π/2, 0), (3π/4, -3), (π, 0)

Example 2: Graph y = 2 cos(x - π/2) + 1

  1. Amplitude: 2
  2. Period: 2π (B=1)
  3. Phase Shift: π/2 to the right
  4. Vertical Shift: 1 unit up
  5. The centerline is y=1. A standard cosine wave starting at a maximum is shifted right by π/2 and oscillates between y=-1 and y=3.

⚠️ Common Mistakes to Avoid

  • Misidentifying the Period: The period is 2π/B, not 2πB. For y=sin(4x), the period is π/2, so the graph cycles faster.
  • Incorrect Phase Shift Direction: For y = sin(B(x - C)), the shift is to the right by C. For y = sin(B(x + C)), it's to the left by C.
  • Forgetting the Vertical Shift: The new maximum is D + A, and the new minimum is D - A. The oscillation happens around y=D, not y=0.

💡 Tips & Tricks

  • Memory Aid: "Cosine is a SINE that's been shifted!" A basic cosine graph is identical to a sine graph shifted left by π/2.
  • Quick Sketch: Always mark the new maximum, minimum, and centerline first. Then divide the period into quarters to place your five key points.
  • Check Your Work: Ensure one full cycle fits exactly within the calculated period length on the x-axis.

🎯 Practice Suggestions

  • Start by graphing simple functions like y = sin(x) and y = cos(x) from -2π to 2π.
  • Practice with one parameter at a time (e.g., only amplitude changes, then only period changes).
  • Use graphing software or a calculator to check your hand-drawn graphs.
  • Try writing the equation from a given graph to reinforce your understanding of the parameters.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 y = 5cos(2(x - π/5)) + 3 = ?

Hint: For a transformed cosine function y = A cos(B(x - C)) + D, identify how each parameter affects the graph. Consider what A, B, C, and D represent in terms of amplitude, period, horizontal shift, and vertical shift.

Show the answer

Answer: Amplitude: 5, Period: π, Phase Shift: π/5 right, Vertical Shift: 3 up

  1. Identify the parameters from the equation y = 5cos(2(x - π/5)) + 3 A = 5 (amplitude) B = 2 (affects period) C = π/5 (phase shift) D = 3 (vertical shift)
  2. Calculate the amplitude Amplitude = |A| = |5| = 5
  3. Calculate the period Period = 2π/|B| = 2π/2 = π
  4. Determine the phase shift Phase shift = C = π/5 (since it's (x - π/5), the shift is π/5 units to the right)
  5. Determine the vertical shift Vertical shift = D = 3 (upward)
  6. Final answer Amplitude: 5, Period: π, Phase Shift: π/5 right, Vertical Shift: 3 up

2 y = 7sin(2(x - π/2)) - 3 = ?

Hint: For a sine function in the form y = A sin(B(x - C)) + D, identify how each parameter affects the graph. Consider what A, B, C, and D represent in terms of transformations.

Show the answer

Answer: Amplitude: 7, Period: π, Phase Shift: π/2 right, Vertical Shift: 3 down

  1. Identify the amplitude (A) The amplitude is the absolute value of the coefficient of sine: |7| = 7
  2. Identify the period Period = 2π/B where B = 2 Period = 2π/2 = π
  3. Identify the phase shift (C) Phase shift = C = π/2 Since it's (x - π/2), the shift is π/2 units to the right
  4. Identify the vertical shift (D) D = -3, so the graph is shifted 3 units downward
  5. Final parameters Amplitude: 7 Period: π Phase Shift: π/2 right Vertical Shift: 3 down

3 y = 8cos(2(x - π/4)) - 6 = ?

Hint: Compare the given equation to the standard form y = A cos(B(x - C)) + D. Each parameter controls a different transformation: A affects the height, B affects the width, C shifts left or right, and D shifts up or down. Pay attention to the sign of D and the direction of the shift in C.

Show the answer

Answer: Amplitude: 8, Period: π, Phase Shift: π/4 right, Vertical Shift: 6 down

  1. Write the equation in standard form: y = 8cos(2(x - π/4)) - 6. This matches y = A cos(B(x - C)) + D.
  2. Identify the amplitude (A). The coefficient of cosine is 8, so amplitude = |8| = 8.
  3. Identify the period. B = 2, so period = 2π / B = 2π / 2 = π.
  4. Identify the phase shift (C). Inside the parentheses we have (x - π/4), so C = π/4. Since it is x minus C, the shift is π/4 units to the right.
  5. Identify the vertical shift (D). D = -6, so the graph is shifted 6 units downward.
  6. Final answer: Amplitude = 8, Period = π, Phase Shift = π/4 right, Vertical Shift = 6 down.

4 y = 10cos(3(x - π/6)) - 4 = ?

Hint: Compare the given equation to the standard form y = A cos(B(x - C)) + D. Each parameter A, B, C, and D controls a specific transformation of the basic cosine graph. Think about what each one does: A stretches vertically, B affects the horizontal stretch (period), C shifts left/right, and D shifts up/down.

Show the answer

Answer: Amplitude: 10, Period: 2π/3, Phase Shift: π/6 right, Vertical Shift: 4 down

  1. Write the equation in standard form: y = 10cos(3(x - π/6)) - 4. This matches y = A cos(B(x - C)) + D.
  2. Identify the amplitude (A). The amplitude is the absolute value of the coefficient of cosine: |10| = 10.
  3. Identify the period. The period is given by 2π/B, where B = 3. So period = 2π/3.
  4. Identify the phase shift (C). The expression inside the cosine is (x - π/6), so C = π/6. Since it is (x - C), the shift is π/6 units to the right.
  5. Identify the vertical shift (D). D = -4, so the graph is shifted 4 units downward.
  6. Final answer: Amplitude = 10, Period = 2π/3, Phase Shift = π/6 right, Vertical Shift = 4 down.

5 y = -6cos(4(x - π/8)) + 2 = ?

Hint: Compare the given equation to the standard form y = A cos(B(x - C)) + D. The amplitude is the absolute value of A. The period is 2π divided by B. The phase shift is determined by C, and the vertical shift is D. Pay attention to the negative sign in front of A.

Show the answer

Answer: Amplitude: 6, Period: π/2, Phase Shift: π/8 right, Vertical Shift: 2 up

  1. Write the equation in standard form: y = -6cos(4(x - π/8)) + 2
  2. Identify the amplitude (A): The coefficient of cosine is -6, so amplitude = |-6| = 6
  3. Identify the period: B = 4, so period = 2π/B = 2π/4 = π/2
  4. Identify the phase shift (C): C = π/8, and since it is (x - π/8), the phase shift is π/8 units to the right
  5. Identify the vertical shift (D): D = 2, so the graph is shifted 2 units upward
  6. Final answer: Amplitude = 6, Period = π/2, Phase Shift = π/8 right, Vertical Shift = 2 up

6 y = 12cos(5(x - π/10)) - 4 = ?

Hint: Compare the given equation to the standard form y = A cos(B(x - C)) + D. Each parameter controls a specific transformation: A changes the height, B changes the width, C moves the graph left or right, and D moves it up or down. Pay attention to the sign of D and the direction of the shift indicated by (x - C).

Show the answer

Answer: Amplitude: 12, Period: 2π/5, Phase Shift: π/10 right, Vertical Shift: 4 down

  1. Write the equation in standard form: y = 12cos(5(x - π/10)) - 4
  2. Identify the amplitude (A): The coefficient of cosine is 12, so amplitude = |12| = 12. This means the graph stretches vertically by a factor of 12.
  3. Identify the period: The coefficient B = 5. Period = 2π/B = 2π/5. This means the graph completes one full cycle in 2π/5 units horizontally.
  4. Identify the phase shift (C): C = π/10. Since the equation is in the form (x - C), the phase shift is π/10 units to the right.
  5. Identify the vertical shift (D): D = -4. This means the graph is shifted 4 units downward.
  6. Final answer: Amplitude = 12, Period = 2π/5, Phase Shift = π/10 right, Vertical Shift = 4 down.
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