- Identify the standard form of a horizontal shift for a function \(f(x)\) as \(f(x - c)\), where \(c\) represents the shift amount.
- Compare the given function \(\cos(x + \frac{\pi}{2})\) with the standard form \(\cos(x - c)\) to find \(c = -\frac{\pi}{2}\).
- Interpret the negative value of \(c\) as a shift to the left by \(\frac{\pi}{2}\) units.
- The correct transformation is \(\boxed{\frac{\pi}{2} \text{ units to the left}}\).
Explanation
- Analyzing the Transformation Rule
We are given the parent function \(y = \cos(x)\) and a transformed function \(y = \cos(x + \frac{\pi}{2})\). Our goal is to determine the horizontal transformation that maps the parent function to the transformed one. A horizontal transformation of a function \(f(x)\) in the form \(f(x - c)\) indicates a shift of \(c\) units horizontally. If \(c > 0\), the shift is to the right by \(c\) units; if \(c < 0\), the shift is to the left by \(|c|\) units.
- Identifying the Shift Value and Direction
We need to write the given function \(y = \cos(x + \frac{\pi}{2})\) in the standard form \(f(x - c) = \cos(x - c)\) to identify the value of \(c\). By comparing the two expressions:
\[x - c = x + \frac{\pi}{2}\]
Subtracting \(x\) from both sides gives:
\[-c = \frac{\pi}{2}\]
Multiplying by \(-1\) yields:
\[c = -\frac{\pi}{2}\]
Since \(c = -\frac{\pi}{2} < 0\), this corresponds to a horizontal shift to the left by \(\left|-\frac{\pi}{2}\right| = \frac{\pi}{2}\) units.
- Verification and Conclusion
To confirm, consider a point on the graph of \(y = \cos(x)\). At \(x = 0\), \(y = \cos(0) = 1\). For the function \(y = \cos(x + \frac{\pi}{2})\), to obtain the same value of \(y = 1\), we must set the argument to zero:
\[x + \frac{\pi}{2} = 0 \implies x = -\frac{\pi}{2}\]
The point \((0, 1)\) on the original function moves to the point \((-\frac{\pi}{2}, 1)\) on the transformed function. This confirms that the entire graph has been shifted \(\frac{\pi}{2}\) units to the left. The correct option is the first one.
Examples
Horizontal shifts are essential in analyzing periodic phenomena such as sound waves or alternating current. For example, if two speakers emit the same sound tone, but one is slightly delayed relative to the other, the resulting wave functions can be represented by a horizontal phase shift. By calculating this shift, engineers can synchronize audio signals to ensure clear, coherent sound reproduction. This is a practical application of understanding how modifying the argument of a periodic function affects its position on the horizontal axis.