To correctly complete the sentence:

The graph of $y = \cos\left( x - \frac{\pi}{2} \right)$ is obtained by shifting the graph of $y = \cos(x)$ $\frac{\pi}{2}$ units to the right.

Explanation:

When you have a function $y = \cos(x - h)$, the graph of the cosine function shifts horizontally by $h$ units. If $h$ is positive (as in $x - \frac{\pi}{2}$), the graph shifts right. If it were $x + h$, it would shift left.

In this case, since it’s $x - \frac{\pi}{2}$, the shift is $\frac{\pi}{2}$ units to the right.

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Would you like further details or explanations on shifts in trigonometric graphs?

5 Related Questions:

1. What happens to the graph of $\sin(x)$ if we shift it by $\pi/2$ units to the left?
2. How does the amplitude of the cosine function change with coefficients like $A \cdot \cos(x)$?
3. What is the effect of a vertical shift on $\cos(x)$?
4. How can we express the phase shift of $y = \cos(x - \pi/3)$?
5. What is the period of a general cosine function $y = \cos(Bx - C)$?

Tip:

Horizontal shifts are always applied in the opposite direction of what they appear: $(x - h)$ shifts right, while $(x + h)$ shifts left!