Problem 62 - Precalculus: A Concise Course

Question

\(g\) is related to a parent function \(f(x)=\sin (x)\) or \(f(x)=\cos (x)\) (a) Describe the sequence of transformations from \(f\) to \(g\). (b) Sketch the graph of \(g\). (c) Use function notation to write \(g\) in terms of \(f\). $$ g(x)=\sin (2 x+\pi) $$

Step-by-Step Solution

Answer

The function \(g(x) = \sin(2x + \pi)\) is a transformation from the function \(f(x) = \sin(x)\). The transformations are a horizontal compression by \(1/2\) and a shift to the left by \(\pi\) units. In sketching the graph, these transformations should be applied to the basic sine function. The function \(g(x)\) can be written in terms of \(f(x)\) as \(g(x) = f(2x + \pi)\).

Step 1: Identify the parent function

The parent function here is \(f(x) = \sin(x)\) because \(g(x)\) consists of sine function. Therefore, the value of \(f(x)\) should be compared with the value of \(g(x)\) to identify the transformations.

Step 2: Identify the transformations

The function \(g(x)\) is a transformation of the function \(f(x)\) by a horizontal compression by a factor of \(1/2\) and a phase shift to the left by \(\pi\) units. The factor of 2 in the argument of the sine function decreases the period of the function, while the addition of \(\pi\) shifts the graph to the left. So, the sequence of transformations is a horizontal compression by \(1/2\) followed by a shift to the left by \(\pi\) units.

Step 3: Sketch the graph

To sketch the graph of \(g(x)\), start with the graph of \(f(x)\). This is a wave starting from the origin, peaking at \(y=1\) at \(x = \pi/2\), going back to \(y=0\) at \(x = \pi\), reaching at \(y=-1\) at \(x = 3\pi/2\) and again going back to \(y=0\) at \(x = 2\pi\). Apply the transformations to this basic shape of sine function. Start by reducing the period by half - this compresses the graph horizontally. Then, shift the graph to the left by \(\pi\) units, moving the start of the wave from the origin to \(x = -\pi\). Keep all the peaks, troughs and zeros in line with the parent function \(f(x)\), but shifted and compressed accordingly.

Step 4: Write \(g\) in terms of \(f\)

The function \(g\) can be written in terms of the function \(f\) as follows: \(g(x) = f(2x + \pi)\). This notation tells us that the function \(g\) is generated by applying the transformations (horizontal compression by \(1/2\) and shift to the left by \(\pi\) units) to the function \(f\).

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function, which is periodic in nature. This periodic behavior means it repeats its values in a consistent pattern over regular intervals, known as periods. In the parent sine function, which is denoted as \( f(x) = \sin(x) \), the wave starts its cycle at the origin.

The sine wave has specific characteristics:

• It peaks at 1, which is its maximum value, and troughs at -1, its minimum value.
• The sine function is symmetric and has a period of \(2\pi\); its values repeat every \(2\pi\) units along the x-axis.
• At \(x = 0\), the sine value is 0.

This consistency makes it easy to predict values at any point along the x-axis.

Understanding this basic format helps when manipulating or transforming sine functions.

Horizontal Compression

Horizontal compression is one of the transformations that can alter the appearance of a graph, by changing its period. For the function \(g(x) = \sin(2x + \pi)\), a horizontal compression occurs.

When there's a multiplier inside the sine function, as seen with the 2 in \(\sin(2x)\), it affects the period of the graph:

• A horizontal compression by a factor happens when we multiply the x-value inside the function by a number greater than 1.
• The original period of \(2\pi\) is compressed to \(\frac{2\pi}{b}\), where \(b\) is the multiplier. So in our case, the new period becomes \(\pi\).

Remember, for a sine function, multiplying inside results in compressing or stretching along the x-axis.

Phase Shift

Phase shift describes the horizontal movement of the graph along the x-axis. In \(g(x) = \sin(2x + \pi)\), a phase shift occurs due to the \(+\pi\) in the expression.

Phase shifts follow these rules:

• If you add a constant inside the function, you move the graph to the left by that amount.
• If you subtract a constant, the graph moves to the right.
• In our function, the expression \(2x + \pi\) shifts the entire sine wave to the left by \(\pi\) units.

This shift changes the starting point of the wave but retains its shape and periodic nature.

This transformation is consistent across trigonometric functions, making it a useful tool when transforming graphs for desired phase outcomes.