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. $$ y=\frac{9}{5} \cos \left(-\frac{3 \pi}{2} x\right) $$

Short answer

Domain: \( (-\infty, \infty) \); Range: \( \left[-\frac{9}{5}, \frac{9}{5}\right] \)

Step-by-step solution

Identify the basic function

The given function is based on the cosine function. The basic function is: \[ y = \frac{9}{5} \cos \left(-\frac{3\pi}{2} x\right) \]

Recognize the amplitude

The amplitude is the coefficient of the cosine function, which is \( \frac{9}{5} \). This means the graph will oscillate between \( \frac{9}{5} \) and \( -\frac{9}{5} \).

Determine the period

The period of a cosine function \( y = A \cos (Bx) \) is given by \( \frac{2\pi}{|B|} \). Here, \( B = -\frac{3\pi}{2} \). The period is: \[ \frac{2\pi}{\left|-\frac{3\pi}{2}\right|} = \frac{2\pi}{\frac{3\pi}{2}} = \frac{2\pi \cdot 2}{3\pi} = \frac{4}{3} \]

Determine the phase shift

The phase shift of a cosine function is given by \( -\frac{C}{B} \), where the function is in the form \( y = A \cos (Bx - C) \). In this case, there is no phase shift as the function does not have a \( -C \) term.

Vertical shift

There is no vertical shift as there are no added or subtracted constants in the function.

Identify key points of one period

The key points of the basic cosine function are: \( (0, 1) \), \( (\frac{\pi}{2}, 0) \), \( (\pi, -1) \), \( (\frac{3\pi}{2}, 0) \), \( (2\pi, 1) \). Multiplying each y-coordinate by the amplitude \( \frac{9}{5} \) and accounting for the negative sign, the key points are: \( (0, \frac{9}{5}) \), \( (\frac{2}{3}, 0) \), \( (\frac{4}{3}, -\frac{9}{5}) \), \( (\frac{2 \cdot 2}{3}, 0) \), \( \frac{8}{3}, \frac{9}{5}) \).

Repeat key points for two cycles

Repeat the key points determined in Step 6 for the next cycle to show a full graph of two cycles.

Determine domain and range

The domain of the cosine function is all real numbers, or \( (-\infty, \infty) \). The range is the set of y-values covered by the function. Based on the amplitude, the range is \( \left[-\frac{9}{5}, \frac{9}{5}\right] \).

Key concepts

cosine function

The cosine function is one of the primary trigonometric functions, symbolized as cos(x). It is particularly useful in representing oscillatory and wave-like behaviors. In its basic form, the cosine function is expressed as: y = cos(x)The graph of the basic cosine function (y = cos(x)) is a smooth, wave-like curve that oscillates between 1 and -1. It starts at the maximum value of 1 when x = 0, descends to 0 at \(\pi/2\), reaches -1 at \(\pi\), rises back to 0 at \(3\pi/2\), and completes the cycle back at 1 by \(2\pi\). This wave pattern repeats infinitely in both directions.

amplitude

The amplitude of a trigonometric function such as the cosine function defines the height of its peaks and the depth of its troughs from the midline (y = 0). It is represented by the coefficient A in the function y = A cos(Bx - C). For the given function:$$ y = \frac{9}{5} \cos \left(-\frac{3\pi}{2} x\right) $$The amplitude is \(\frac{9}{5}\), which means the graph will oscillate between \(\frac{9}{5}\) and \(-\frac{9}{5}\). This coefficient directly affects how tall or short the waves of the graph will appear, providing a visual representation of how intense the oscillations are.

period

The period of a trigonometric function is the horizontal length required for the function to complete one full cycle of its wave pattern. For the cosine function y = A cos(Bx - C), the period (P) is calculated using the formula:$$ P = \frac{2\pi}{|B|} $$In our function, we have:$$ y = \frac{9}{5} \cos \left( -\frac{3\pi}{2} x \right) $$Here, B is \(-\frac{3\pi}{2}\), so:$$ P = \frac{2\pi}{\left| -\frac{3\pi}{2} \right|} = \frac{2\pi}{\frac{3\pi}{2}} = \frac{4}{3} $$Thus, the period of our cosine function is \(\frac{4}{3}\), meaning it will repeat its pattern every \(\frac{4}{3}\) units along the x-axis.

phase shift

Phase shift refers to the horizontal displacement of the graph of a trigonometric function from its standard position. For a function in the form y = A cos(Bx - C), the phase shift is calculated as:$$ \text{Phase Shift} = -\frac{C}{B} $$In the given function:$$ y = \frac{9}{5} \cos \left( -\frac{3\pi}{2} x \right) $$There is no C term, meaning there is no horizontal translation applied to the function. Therefore, the phase shift is effectively zero, and the function's graph will not be displaced horizontally. It will start its cycle at the origin, just like the basic cosine function.

trigonometric transformations

Trigonometric transformations involve modifying the basic trigonometric functions (sine, cosine, tangent, etc.) to create shifts, stretches, and reflections. These transformations allow us to adjust the graph's shape and position. Key types of transformations include:

• **Amplitude Change**: Alters the height of the wave. In our function \(\frac{9}{5}\) changes the amplitude.
• **Period Change**: Modifies the length of one complete wave cycle. The period for \(y = \frac{9}{5} \text{cos} (\frac{3\text{pi}}{2} x)\) is \(\frac{4}{3}\).
• **Phase Shift**: Represents horizontal displacement, which is not applicable in our function.
• **Vertical Shift**: Moves the graph up or down. Our function has no vertical shift.

These transformations help in fine-tuning the graph to fit specific requirements or modeling scenarios.