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. $$ y=-\frac{3}{2} \cos \left(\frac{\pi}{4} x\right)+\frac{1}{2} $$

Short answer

The domain is \( (-\infty, \infty) \) and the range is \( [-1, 2] \), graphed with key points shifted and reflected to match transformations.

Step-by-step solution

Identify the Basic Function

The basic function is the cosine function: \( y = \cos(x) \).

Apply the Transformations

To graph \( y = -\frac{3}{2} \cos(\frac{\pi}{4} x) + \frac{1}{2} \), recognize the transformations involved:1. The amplitude is given by the coefficient \( \frac{3}{2} \).2. The negative sign outside the cosine function indicates a reflection across the x-axis.3. The period is adjusted by the \( \frac{\pi}{4} \) coefficient, calculated as \( \frac{2\pi}{\frac{\pi}{4}} = 8 \).4. The vertical shift is given by the addition of \( \frac{1}{2} \).

Determine Key Points

Key points of the cosine function within one period (normally 0 to \( 2\pi \)) are at 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). These points transform as follows:1. For \( x = 0 \), \( y = -\frac{3}{2}\cos(0) + \frac{1}{2} = -\frac{3}{2}(1) + \frac{1}{2} = -1 \).2. For \( x = 8 \), \( y = -\frac{3}{2}\cos(\pi) + \frac{1}{2} = -\frac{3}{2}(-1) + \frac{1}{2} = 2 \).3. For \( x = 4 \), \( y = -\frac{3}{2}\cos(\frac{\pi}{2}) + \frac{1}{2} = -\frac{3}{2}(0) + \frac{1}{2} = \frac{1}{2} \).4. For \( x = 12 \), \( y = -\frac{3}{2}\cos(\frac{3\pi}{2}) + \frac{1}{2} = -\frac{3}{2}(0) + \frac{1}{2} = \frac{1}{2} \).

Plot the Points and Graph Two Cycles

Graph the points (0, -1), (4, \( \frac{1}{2} \)), (8, 2), (12, \( \frac{1}{2} \)), and (16, -1). Then draw the smooth cosine wave through two cycles, noting the amplitude and vertical shift.

Determine the Domain and Range

The domain of the function is all real numbers: \( (-\infty, \infty) \). The range is determined by the highest and lowest values of \( y \), which are -1.5 and 2, respectively. Therefore, the range is \( [-1, 2] \).

Key concepts

cosine function

The cosine function is one of the primary trigonometric functions, essential for understanding waves and oscillations. Its basic form is written as: \[ y = \cos(x) \]. This basic cosine wave oscillates between -1 and 1, creating a smooth, periodic curve. Its key points within one period (from 0 to \( 2\pi \)) are: 0, \( \frac{\pi}{2} \), \( \pi \), \( \frac{3\pi}{2} \), and \( 2\pi \). These points mark the wave's peaks, troughs, and zeroes. Understanding these key points helps in transforming and graphing more complex trigonometric functions.

function transformations

Transforming a function involves changing its graph's shape, position, or orientation. For the given function \( y = -\frac{3}{2} \cos(\frac{\pi}{4} x ) + \frac{1}{2} \), several transformations occur:

• Amplitude Change: The coefficient \( \frac{3}{2} \) scales the cosine wave vertically. It stretches the wave, making the peaks and troughs more pronounced.
• Reflection: The negative sign before the cosine function reflects the wave across the x-axis, flipping it upside down.
• Period Adjustment: The \( \frac{\pi}{4} \) factor inside the cosine function modifies its period. Instead of the usual \( 2\pi \) period, it becomes \( \frac{2\pi }{\frac{\pi }{4}} = 8 \).
• Vertical Shift: Adding \( \frac{1}{2} \) shifts the entire graph upward by 0.5 units. This affects the wave's midline, moving it from y = 0 to y = 0.5.

By applying these transformations, we can plot specific points and create a more complex, yet understandable graph.

amplitude and period

The amplitude and period are crucial characteristics of trigonometric functions:

• Amplitude: The amplitude represents the wave's height from its midline to its peak or trough. For the function \( y = -\frac{3}{2} \cos(\frac{\pi}{4} x ) + \frac{1}{2} \), the amplitude is \( \frac{3}{2} \). This means the wave oscillates 1.5 units above and below its midline.
• Period: The period indicates the duration it takes for the wave to complete one full cycle. Normally, a cosine wave has a period of \( 2\pi \). Due to the transformation factor \( \frac{\pi}{4} \), our function's period becomes 8. This change stretches the wave horizontally, making each cycle cover an interval of 8 units.

Identifying these two properties helps in comprehending and accurately graphing trigonometric functions with transformations.

domain and range

Defining a function's domain and range helps understand its behavior:

• Domain: The domain encompasses all possible x-values a function can take. For the cosine function and its transformations, the domain remains all real numbers: \( (-\infty, \infty) \). This indicates the wave extends infinitely in both directions, covering every x-value.
• Range: The range specifies the y-values the function can output. After applying all transformations to our function, the range is \( [-1, 2] \). The lowest point, -1, occurs at the wave's trough, while the highest point, 2, occurs at the peak.

Understanding the domain and range of a transformed function provides a complete picture of its behavior, aiding in accurate graphing and interpretation.