Amplitude, Period, Range, and Phase Shift of Trigonometric Functions

Practice determining the amplitude, period, range, and phase shift of trigonometric functions. Multiple-choice questions are followed by detailed step-by-step solutions.

Practice Questions

Question 1

If \( y = \cos x \), what is the maximum value of \( y \)?
  1. 1
  2. -1
  3. \(\pi\)
  4. \(2\pi\)

Question 2

What is the period of \( f(x) = 2\sin(5x) \)?
  1. \(\pi/5\)
  2. \(2\pi/5\)
  3. \(5\pi\)
  4. \(\pi\)

Question 3

What is the amplitude of \( f(x) = -3\cos(\pi x) \)?
  1. 3
  2. -3
  3. \(\pi\)
  4. 2

Question 4

Which of the following functions has the greatest period?
  1. \(20\sin(2x-\pi/2)\)
  2. \(-\sin(\pi x)\)
  3. \(2\sin(0.1x)\)
  4. \(-\sin(0.1\pi x)\)

Question 5

What is the range of \( f(x) = -4\cos(2x-3) \)?
  1. \((0,4)\)
  2. \([0,4]\)
  3. \((-4,4)\)
  4. \([-4,4]\)

Question 6

What is the phase shift of \( f(x) = 7\sin(2x-\pi/3) \)?
  1. \(\pi/3\)
  2. \(\pi/6\)
  3. \(-\pi/6\)
  4. \(-\pi/3\)

Question 7

What is the range of \( f(x) = -6\cos(\pi x-\pi/2)+2 \)?
  1. \([-6,6]\)
  2. \([-4,8]\)
  3. \([0,8]\)
  4. \([-6,0]\)

Question 8

What is the amplitude of \( f(x) = 4\sin x \cos x \)?
  1. 4
  2. 3
  3. 2
  4. 1

Question 9

What is the period of \( f(x) = 0.5\sin x \cos x \)?
  1. 0.5
  2. \(2\pi\)
  3. \(\pi/2\)
  4. \(\pi\)

Question 10

What is the amplitude of \( f(x) = \sin x + \cos x \)?
  1. \(\sqrt{2}\)
  2. \(\sqrt{2}/2\)
  3. \(2\sqrt{2}\)
  4. 2

Step-by-Step Solutions

  1. Q1: The cosine function satisfies \( -1 \le \cos x \le 1 \). Maximum value: \(1\).
  2. Q2: For \( \sin(bx) \), the period is \( \frac{2\pi}{b} \). Here \( b = 5 \), so \[ T = \frac{2\pi}{5} \]
  3. Q3: Amplitude is the absolute value of the coefficient: \[ | -3 | = 3 \]
  4. Q4: Period comparison: Largest period: \(2\sin(0.1x)\).
  5. Q5: Base range of cosine: \([-1,1]\). Multiply by 4: \[ [-4,4] \]
  6. Q6: Phase shift formula: \[ \text{Phase shift} = \frac{c}{b} \] Here \( b=2 \), \( c=\pi/3 \): \[ \frac{\pi/3}{2} = \frac{\pi}{6} \]
  7. Q7: Amplitude: 6 → range \([-6,6]\). Vertical shift: +2: \[ [-6+2,6+2] = [-4,8] \]
  8. Q8: Use identity: \[ \sin x \cos x = \frac{1}{2}\sin(2x) \] So: \[ f(x) = 2\sin(2x) \] Amplitude = 2.
  9. Q9: \[ 0.5\sin x \cos x = 0.25\sin(2x) \] Period of \( \sin(2x) \) is: \[ \pi \]
  10. Q10: Rewrite: \[ \sin x + \cos x = \sqrt{2}\sin\left(x+\frac{\pi}{4}\right) \] Amplitude = \( \sqrt{2} \).

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