Question

Beginning with the graphs of \(y=\sin x\) or \(y=\cos x,\) use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility only to check your work. $$f(x)=3 \sin 2 x$$

Short answer

Question: Sketch the graph of the function \(f(x)=3\sin 2x\). Describe the steps you took to create the graph. Answer: I began by sketching the basic sine wave, \(y=\sin x\), which oscillates between 1 and -1 with a period of \(2\pi\). Then, I applied a vertical scaling by a factor of 3, so the new maximum and minimum values were 3 and -3. Next, I applied a horizontal scaling by a factor of 1/2, resulting in a new period of \(\pi\). After combining both scaling transformations, the final graph represents the function \(f(x) = 3 \sin 2x\) with a maximum value of 3, a minimum value of -3, and a period of \(\pi\). I verified the accuracy of my sketch using a graphing utility.

Step-by-step solution

Sketch the basic sine wave

Begin by sketching the basic sine wave, \(y=\sin x\). The sine wave has a maximum value of 1, a minimum value of -1, and a period of \(2\pi\). It oscillates between these maximum and minimum values as \(x\) ranges from \(0\) to \(2\pi\) and repeats itself every \(2\pi\).

Apply a vertical scaling by a factor of 3

Now, apply a vertical scaling to the sine wave by a factor of 3. This means the amplitude or maximum and minimum values of the wave will be multiplied by 3. As a result, the new maximum value will be 3 and the new minimum value will be -3. Modify the graph of the sine wave so that it oscillates between these new maximum and minimum values.

Apply a horizontal scaling by a factor of 1/2

Next, apply a horizontal scaling by a factor of 1/2, meaning that every \(x\)-value will be multiplied by 1/2. Since the sine wave has a period of \(2\pi\), this scaling will cause the new period to be \(2\pi \times (1/2) = \pi\). This means the wave now completes one full oscillation from maximum value to minimum value and back within the range of \(0\) to \(\pi\). Adjust the graph so that it reflects this new period.

Combine the scaling transformations to obtain the final graph

With both vertical and horizontal scalings applied, the graph should now represent the function \(f(x) = 3 \sin 2x\). The wave should have a maximum value of 3, a minimum value of -3, and a period of \(\pi\).

Check your work with a graphing utility

Finally, use a graphing utility to confirm that your sketch is accurate. Plot the function \(f(x) = 3\sin 2x\) and verify that the graph on the screen matches your sketched graph with the scaling transformations applied.

Key concepts

Sine Function

The sine function, denoted as \( y = \sin x \), is one of the fundamental trigonometric functions. It's defined on the interval from \( -\infty \) to \( \infty \) and exhibits a continuous oscillatory pattern. This function creates a wavelike pattern known as a sine wave, characterized by:

• A smooth and periodic wave pattern.
• A range of values from -1 to 1.
• Repeats every \( 2\pi \), which is the period.

The basic sine wave starts at zero, rises to a maximum, descends to zero, plunges to a minimum, and then returns to zero, completing one full cycle. This simplistic elegance forms the basis of understanding sine transformations, essential for modifying its amplitude and period according to other function parameters.

Amplitude

Amplitude refers to the peak value the sine function reaches from its equilibrium position, or middle line. Initially, \( y = \sin x \) oscillates between the maximum of 1 and minimum of -1, implying an amplitude of 1.

• High amplitude translates to taller waves, indicating more extreme peaks and troughs.
• Low amplitude leads to shorter, flatter waves.

When we encounter functions like \( f(x) = 3 \sin 2x \), the number 3 before the sine function causes a vertical scaling, effectively stretching the wave. Here, the amplitude triples: the maximum reaches 3, and the minimum hits -3. This scaling doesn't affect the wave's frequency but modifies the height of oscillations, enhancing its vertical dimension.

Period

The period of a sine function signifies the horizontal length required for one complete cycle. For the basic sine function \( y = \sin x \), this is \( 2\pi \). In the function \( f(x) = 3 \sin 2x \), the 2 within \( \sin 2x \) causes a horizontal scaling. This directly affects the period. Here's how it works:

• The period formula is \( \frac{2\pi}{b} \), where \( b \) is the horizontal scaling factor, here \( 2 \).
• Replacing \( b \) with 2, the new period is \( \frac{2\pi}{2} = \pi \).

Thus, the sine wave compresses horizontally. It now completes its cycle in a shorter distance, \( \pi \) instead of \( 2\pi \), resulting in a steeper frequency. Recognizing period changes is crucial for understanding how wave speed and intervals shift in transformed functions.

Scaling Transformations

Scaling transformations involve resizing the sine wave either vertically or horizontally, impacting its amplitude or period respectively. Vertical transformations are straightforward:

• Multiplying the sine function by a constant increases its amplitude. If the constant is negative, the wave also flips.
• This doesn't affect the period but alters how high or low the wave reaches.

Horizontal transformations change the frequency:

• A multiplier inside \( \sin(2x) \) modifies the period. When \( b > 1 \), as in our example, it shortens the period, \( \frac{2\pi}{b} \).
• Conversely, if \( 0 < b < 1 \), it lengthens the period, extending the wave's horizontal stretch.

Combined scaling leads to complex yet predictable waveforms. Applying both vertical and horizontal scalings produces transformations present in \( f(x) = 3 \sin 2x \). With practice, predicting these alterations becomes more intuitive, vastly enhancing analysis and graph skills.