Which of the following functions illustrates a change in amplitude? A. y = 3cos(4x) B. y = 1 + sin(x) C. y = -2 – cos(x – Ï€) D. y = tan(2x)
The Correct Answer and Explanation is:
Explanation
In trigonometric functions, amplitude refers to the maximum vertical distance from the midline (or equilibrium) of the graph to its peak or trough. For sine and cosine functions, the general form is:y=A⋅sin(Bx+C)+Dory=A⋅cos(Bx+C)+Dy = A \cdot \sin(Bx + C) + D \quad \text{or} \quad y = A \cdot \cos(Bx + C) + Dy=A⋅sin(Bx+C)+Dory=A⋅cos(Bx+C)+D
Here, |A| represents the amplitude of the wave. If A = 1, the function oscillates between -1 and 1. If A = 3, the function stretches vertically to oscillate between -3 and 3.
Let’s examine each option:
- A. y = 3cos(4x)
This function has A = 3, so the amplitude is 3. This means the graph is a vertically stretched cosine wave compared to the standard cos(x), which has an amplitude of 1.
✅ This function shows a change in amplitude. - B. y = 1 + sin(x)
This is a vertical translation of the sine wave. The graph is shifted up by 1 unit, but the amplitude remains 1.
❌ No change in amplitude. - C. y = -2 – cos(x – π)
This function has a vertical reflection and downward shift, but the amplitude is determined by the coefficient of cosine, which is still 1 (since there’s no number multiplied directly with cos). The “-2” is a vertical translation.
❌ No change in amplitude. - D. y = tan(2x)
The tangent function does not have an amplitude because it is unbounded — its values go to positive and negative infinity.
❌ Amplitude is not defined for tangent.
Conclusion: Only option A shows a change in amplitude due to the coefficient 3 multiplying the cosine function.
