Match each function with its graph in choices A–I. (One choice will not be used.)
y = sin (x - π/4)


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D. <IMAGE> E. <IMAGE> F. <IMAGE>


G. <IMAGE> H. <IMAGE> I. <IMAGE>

Verified step by step guidance

1. Recall that the function given is \(y = \sin\left(x - \frac{\pi}{4}\right)\), which represents a horizontal shift of the basic sine function \(y = \sin x\).
2. Understand that the expression \(x - \frac{\pi}{4}\) inside the sine function means the graph of \(y = \sin x\) is shifted to the right by \(\frac{\pi}{4}\) units.
3. Identify key points on the basic sine curve, such as zeros at multiples of \(\pi\), maximum at \(\frac{\pi}{2}\), and minimum at \(\frac{3\pi}{2}\), then shift all these points to the right by \(\frac{\pi}{4}\) to predict the shape of the graph for \(y = \sin\left(x - \frac{\pi}{4}\right)\).
4. Compare the predicted shifted sine wave with each graph option (A through I), looking for the graph that matches the rightward shift by \(\frac{\pi}{4}\) without any changes in amplitude or period.
5. Select the graph that correctly shows the sine wave shifted right by \(\frac{\pi}{4}\) and ignore the one choice that does not correspond to any function given.

Key Concepts

Sine Function and Its Properties

The sine function, sin(x), is a periodic wave with amplitude 1, period 2π, and zero crossings at multiples of π. Understanding its shape, maximum and minimum values, and where it crosses the x-axis is essential for identifying its graph.

Phase Shift in Trigonometric Functions

A phase shift occurs when the input variable x is replaced by (x - c), shifting the graph horizontally by c units. For y = sin(x - π/4), the sine wave shifts π/4 units to the right, altering the position of peaks, troughs, and zeros without changing amplitude or period.

Graph Matching Techniques

Matching a function to its graph involves analyzing key features like amplitude, period, phase shift, and intercepts. Recognizing how transformations affect the graph helps eliminate incorrect options and identify the correct match among multiple choices.