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. $$g(x)=-2 \cos (x / 3)$$

Short answer

Question: Sketch the graph of the function \(g(x) = -2 \cos (x / 3)\) based on the graphs of \(y = \sin x\) and \(y = \cos x\). Answer: To sketch the graph of \(g(x) = -2 \cos (x / 3)\), apply the following transformations to the graph of \(y = \cos x\): vertical scaling by a factor of 2, reflection in the x-axis, and horizontal scaling by a factor of 3. The resulting graph will have an amplitude of 2, and it will be reflected and compressed horizontally. Verify the final sketch using a graphing utility if needed.

Step-by-step solution

Identify the transformations

Based on the function \(g(x)=-2 \cos (x / 3)\), we can notice the following transformations: 1. Scaling by a factor of \(2\) in the vertical direction, which will stretch the graph vertically. 2. Negation of the cosine function, which reflects the graph in the x-axis. 3. Scaling by a factor of \(3\) in the horizontal direction, which will compress the graph horizontally.

Draw the original graph and apply the transformations

Start by sketching the original graph of \(y=\cos x\). 1. Vertical scaling: Multiply all y values by 2 to make the amplitude 2. 2. Reflection in x-axis: Negate the y values after the first transformation so that the resulting function is \(-2\cos x\). 3. Horizontal scaling: Divide the x values by 3 for the cosine function giving \(y=-2 \cos (x / 3)\), this will compress the graph horizontally.

Check the final sketch using a graphing utility

Use a graphing utility such as Desmos or a graphing calculator to plot the function \(g(x)=-2 \cos (x / 3)\). This will verify whether the drawn sketch is correct. Compare the drawn graph with the utility's output, and if they match, the sketch is correct. Otherwise, check the applied transformations again and make necessary corrections.

Key concepts

scaling transformations

Scaling transformations are essential in modifying the size of a function's graph without altering its fundamental shape. When we talk about scaling vertically, we refer to changes in the amplitude of the function. Similarly, horizontal scaling affects the period or frequency of a graph.

For the function \(g(x) = -2 \cos (x / 3)\), two key scaling transformations happen:

• Vertical Scaling: The coefficient \(-2\) indicates that the amplitude, or maximum vertical distance from the center line to the peak of the graph, is 2 instead of 1 (which is typical for a standard cosine function). This means you multiply all \(y\) values of the original \(\cos x\) graph by 2.
• Horizontal Scaling: The term \(x / 3\) points to how frequent the oscillations occur. By dividing \(x\) by 3, we compress the graph horizontally by a factor of 3, resulting in waves squeezing together more closely.

These scaling transformations are what morph the simple \(\cos x\) graph into \(g(x) = -2 \cos (x / 3)\). Each section of the graph becomes taller and tighter, but the fundamental wave-like structure remains.

graphing utility

A graphing utility is a digital tool or device that can plot functions to verify and explore their shapes and transformations. Tools like Desmos, GeoGebra, or a computerized graphing calculator allow you to input equations and observe the resulting graphs. This serves as a great method to check the accuracy of manually sketched transformations.

For our exercise with \(g(x) = -2 \cos (x / 3)\), using a graphing utility involves these simple steps:

• Input the transformed function equation into the utility.
• Observe the plotted graph and compare it to the hand-drawn version.
• Adjust any discrepancies by revisiting each transformation step.

With a graphing utility, you gain instant visual feedback and can easily experiment with additional transformations to see how the changes reflect on the graph. It's a powerful way to build confidence and understanding of function transformations.

cosine function

The cosine function, denoted as \(\cos x\), is a trigonometric function representing the cosine of an angle \(x\). It is periodic with a fundamental cycle of \(2\pi\), meaning it repeats every \(2\pi\) units.

In its standard form, \(y = \cos x\) has distinct features:

• Maximum value of 1 and a minimum value of -1, giving it an amplitude of 1.
• A period of \(2\pi\), which tells us each full oscillation occurs over \(2\pi\) units.
• The graph is symmetric about the y-axis (even function).

Understanding \(\cos x\) as the starting point is important for transformations. Scaling and other changes, as we've applied in \(g(x) = -2 \cos (x / 3)\), modify its amplitude, period, and general shape. Recognizing these original properties allows for more intuitive manipulation and insight into how they affect sine and cosine graphs in an applied context.

horizontal compression

Horizontal compression is a transformation that affects the period of a trigonometric function, shortening the distance between each cycle. This is achieved by multiplying the variable \(x\) by a factor greater than 1, typically appearing as a division in transformation expressions.

In \(g(x) = -2 \cos (x / 3)\), the presence of \(x/3\) implies a horizontal scaling:

• The division by 3 means compressing the period of the function. Where \(\cos x\) normally completes a cycle in \(2\pi\), \(\cos(x/3)\) accomplishes the same in \(\frac{2\pi}{3}\).
• Consequently, the waves become more frequent over the same interval, densely packing the hills and troughs of the graph.

This transformation fundamentally alters the horizontal layout of the graph, demonstrating how horizontal compressions adjust the frequency and spacing of periodic functions. Recognizing and applying this concept is crucial for accurately plotting and understanding trigonometric transformations.