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. $$ y=2-4 \cos (3 x) $$

Short answer

Domain: \((-\infty, \infty)\), Range: \([-2, 6]\)

Step-by-step solution

Identify the basic function and its properties

The given function is a transformation of the cosine function: \(y = \,2 - 4 \cos(3x)\). The basic cosine function is \(y = \cos(x)\), which has an amplitude of 1, period of \(2\pi\), and is centered at the y-axis.

Amplitude and vertical shift

In the given function, the amplitude is modified by the coefficient -4. The amplitude is the absolute value of this coefficient, which is 4. The vertical shift is indicated by the constant term, 2. This means the whole graph shifts 2 units up.

Period and frequency

The period of the cosine function is affected by the coefficient of the x-term inside the cosine function. For \(y = \cos(3x)\), the period is calculated as \(\frac{2\pi}{3}\). Thus, the graph will complete one cycle over an interval of \(\frac{2\pi}{3}\).

Identify key points

Identify the key points of a standard cosine function (y-intercept, maximum, x-intercepts, and minimum) and then scale and translate these points: 1. Maximum: Adjust the cosine max from 1 to -4 and then shift up by 2 units: \(\left(0, -4\right) \rightarrow \left(0, 2 + -4\right) = \left(0, -2\right)\) 2. Minimum: Shift the cosine min, \(1 - 4 + 2 = 6\). 3. X-intercepts: Translate and scale the intervals.

Draw the graph

Plot the transformed key points and draw the scaled cosine curve repeating for two cycles. Ensure the amplitude is 4, the vertical shift is 2 up, and the period is \(\frac{2\pi is}{3}\). Label the x-intercepts, maximum, minimum, and midline shifts on the graph.

Determine the domain and range

The domain of the function is all real numbers because cosine function is defined for all real values of x, i.e., \((-\infty, \infty)\). The range is determined using the transformed amplitude and vertical shift: \([-2, 6]\).

Key concepts

Cosine Function

The cosine function, written as \(y = \,\cos(x)\), is a periodic function that creates a wave-like graph. This basic function has key properties:

• Amplitude: The height from the centerline to the peak (maximum) or to the trough (minimum). For the basic cosine function, the amplitude is 1.
• Period: The interval over which the function repeats itself. For \(y = \,\cos(x)\), the period is \(2\pi\).
• Maximum and Minimum Values: For \(y = \,\cos(x)\), the maximum value is 1 and the minimum value is -1.
• Key Points: The key points of \(y = \,\cos(x)\) in one period are (0,1), \(\left(\frac{\pi}{2}, 0\right)\), \(\left(\pi, -1\right)\), \(\left(\frac{3\pi}{2}, 0\right)\), and \(2\pi, 1\).

These properties help in graphing and understanding transformations applied to the basic cosine function.

Function Transformations

Function transformations involve shifting, stretching, or compressing the graph of a function. These transformations change the appearance of the graph but not the type of function it represents.
For the given function \(y = 2 - 4\cos(3x)\), multiple transformations are applied:

• Vertical Shift: The constant 2 means the entire graph moves 2 units up.
• Vertical Stretch and Reflection: The coefficient -4 changes the amplitude and flips the graph vertically. The amplitude becomes |4|, indicating a stretch, and the negative sign reflects it across the x-axis.
• Horizontal Compression: The coefficient 3 inside the cosine function affects the period, compressing the graph horizontally. The period becomes \(\frac{2\pi}{3}\).

Understanding how each transformation affects the graph helps in accurately plotting and analyzing the modified function.

Amplitude and Period

Amplitude and period are crucial in defining the shape and frequency of trigonometric functions like cosine.
Amplitude is the height from the midline (center of the wave) to the peak or trough. For the function \(y = 2 - 4\cos(3x)\), the amplitude is the absolute value of the coefficient: | -4 |, which equals 4. This tells us how high and low the curve reaches from its midline.
Period is the length over which the function repeats itself. For \(y = 2 - 4\cos(3x)\), the period is calculated by dividing \(2\pi\) (the period of the basic cosine function) by the coefficient of x inside the function:
\[ \text{Period} = \frac{2\pi}{3} \]
This means the function completes one full cycle every \(\frac{2\pi}{3}\) units along the x-axis. Knowing the amplitude and period helps in sketching the graph accurately and understanding its behavior.

Domain and Range

The domain and range of a trigonometric function define the set of possible input values (domain) and the set of possible output values (range).
Domain of the cosine function \(y = 2 - 4\cos(3x)\) is all real numbers, \( (-\infty, \infty) \), because cosine is defined for all x-values.
Range is determined by the amplitude and vertical shift:

• Midline: The vertical shift moves the midline up to y = 2.
• Maximum: The highest value reaches 2 (midline) + 4 (amplitude) = 6.
• Minimum: The lowest value reaches 2 (midline) - 4 (amplitude) = -2.

Thus, the range is \-2, 6\. Understanding the domain and range assists in determining the possible values the function can take, which is essential for graphing and solving equations involving trigonometric functions.