Question

Graph each function using transformations or the method of key points. Be sure to label key points and show at least two cycles. Use the graph to determine the domain and the range of each function. $$ y=-3 \cos x $$

Short answer

Domain: \((-\infty, \infty)\), Range: \([-3, 3]\)

Step-by-step solution

Identify the Base Function

The base function here is \(y = \cos x \), which is a cosine wave.

Identify Transformations

The given function is \(y = -3 \cos x \). This involves two transformations: a vertical stretch by a factor of 3 and a reflection across the x-axis.

Determine Key Points

For \(y = \cos x \), key points in one cycle are (0,1), (\( \pi /2 \),0), (\( \pi \),-1), (3\( \pi /2 \),0), and (2\( \pi \),1). After applying the transformation, the points should be adjusted accordingly: (0,-3), (\( \pi /2 \),0), (\( \pi \),3), (3\( \pi /2 \),0), and (2\( \pi \),-3).

Graph the Function

On the x-axis, mark points from 0 to 4\( \pi \) to show two full cycles. Plot the key points identified: (0,-3), (\( \pi /2 \),0), (\( \pi \),3), (3\( \pi /2 \),0), (2\( \pi \),-3), (5\( \pi /2 \),0), (3\( \pi \),3), (7\( \pi /2 \),0), and (4\( \pi \),-3). Draw the cosine wave through these points.

Determine Domain and Range

The domain of the cosine function is all real numbers, thus the domain is \( (-\infty, \infty) \). The range is determined by the vertical stretch and reflection, giving the range \([-3, 3] \).

Key concepts

cosine function

The cosine function, represented as \(y = \cos x\), is one of the fundamental trigonometric functions. It describes a wave-like pattern that oscillates between -1 and 1. This wave repeats every \(2\pi \) radians, which is called the period of the function. The graph of the basic cosine function starts at \((0, 1)\), reaches \((\pi/2, 0)\), dips down to \((\pi, -1)\), returns to \((3\pi/2, 0)\), and completes one cycle at \((2\pi, 1)\). In each cycle, the values of the cosine function shift smoothly and predictably.
This baseline understanding is crucial when applying further transformations or identifying key points.

transformations

Transformations adjust the basic shape and position of the cosine graph. For the given function, \(y = -3 \cos x\), there are two key transformations to note:

• Vertical Stretch: The coefficient 3 stretches the graph vertically by a factor of 3, increasing its peak and trough values from 1 and -1 to 3 and -3, respectively.
• Reflection: The negative sign in front of the cosine function reflects the graph across the x-axis, flipping it upside down. What used to be a peak becomes a trough and vice versa.

Understanding these transformations helps in accurately drawing and analyzing the function.

key points

Key points are specific values that help in plotting the graph of a trigonometric function. For the function \(y = -3 \cos x\), we start with the key points of the basic cosine function and apply the transformations:

• Original key points: \((0, 1), (\pi/2, 0), (\pi, -1), (3\pi/2, 0), (2\pi, 1)\).
• After vertical stretch and reflection: \((0, -3), (\pi/2, 0), (\pi, 3), (3\pi/2, 0), (2\pi, -3)\).

Repeating these transformations for the second cycle, we get: \((2\pi, -3), (5\pi/2, 0), (3\pi, 3), (7\pi/2, 0), (4\pi, -3)\). Plotting these points and drawing a smooth curve through them completes the graph of the transformed function. Key points simplify the process of graphing complex trigonometric functions.

domain and range

The domain and range provide important information about where the function is defined and the possible values it can take. For the function \(y = -3 \cos x\):

• Domain: The cosine function is defined for all real numbers, hence the domain of \(y = -3 \cos x\) is \( (-\infty, \infty) \).
• Range: The vertical stretch by 3 and reflection affect the range. Originally, the cosine function ranges between -1 and 1. With these transformations, the range shifts to \([-3, 3]\).
  
  Knowing the domain and range helps to understand the extent and limits of the function across both axes.