Question

a) Determine the range of each function. i) \(y=3 \cos \left(x-\frac{\pi}{2}\right)+5\) ii) \(y=-2 \sin (x+\pi)-3\) iii) \(y=1.5 \sin x+4\) iv) \(y=\frac{2}{3} \cos \left(x+50^{\circ}\right)+\frac{3}{4}\) b) Describe how to determine the range when given a function of the form \(y=a \cos b(x-c)+d\) or \(y=a \sin b(x-c)+d\).

Short answer

i) [2, 8] ii) [-5, -1] iii) [2.5, 5.5] iv) [0.083, 1.417]. For functions in forms \( \text{y}=\text{a} \cos \left( \text{b} \left( \text{x}-\text{c} \right) \right) + \text{d}\) and \( \text{y}=\text{a} \sin \left( \text{b} \right) + \text{d}\), the range is from \( \text{d}-\text{a} \) to \( \text{d}+\text{a} \).

Step-by-step solution

- Understanding the Cosine and Sine Functions

The base functions, \(\text{cos}(x)\) and \(\text{sin}(x)\), have ranges from \([-1, 1]\). Any transformations will modify this range. For a function of the form \(\text{y}=\text{a} \cos \left(\text{b} \left( \text{x}-\text{c} \right) \right) + \text{d}\) or \(\text{y}=\text{a} \sin \left(\text{b} \left( \text{x}-\text{c} \right) \right) + \text{d}\) the amplitude and vertical shift primarily determine the range.

- Find the Range for i) \(\text{y}=3 \cos \left( \text{x} - \frac{\text{\pi}}{2} \right)+5\)

Here \(\text{a}=3\) and \(\text{d}=5\). The range is given by adding and subtracting the amplitude's value (3) from the vertical shift (5). Hence, the range is: \[ \text{Range} = \left[ 5 - 3, 5 + 3 \right] = \left[ 2, 8 \right]. \]

- Find the Range for ii) \(\text{y}=-2 \sin \left( \text{x} + \text{\pi} \right)-3\)

Here \(\text{a}=-2\) and \(\text{d}=-3\). The range will extend \(\text{a}\) units above and below \(\text{d}\). Thus, the range is: \[ \text{Range} = \left[ -3 - 2, -3 + 2 \right] = \left[ -5, -1 \right]. \]

- Find the Range for iii) \(\text{y}=1.5 \sin \text{x}+4\)

With \(\text{a}=1.5\) and \(\text{d}=4\), the range is determined by the vertical shift plus minus the amplitude. Consequently, the range is: \[ \text{Range} = \left[ 4 - 1.5, 4 + 1.5 \right] = \left[ 2.5, 5.5 \right]. \]

- Find the Range for iv) \(\text{y}=\frac{2}{3} \cos \left(\text{x}+50^{\circ}\right)+\frac{3}{4}\)

Checking the values of \(\text{a}=\frac{2}{3}\) and \(\text{d}=\frac{3}{4}\), the range will be centered around \(\frac{3}{4}\), modified by the amplitude \(\frac{2}{3}\). The range is: \[ \text{Range} = \left[\frac{3}{4} - \frac{2}{3}, \frac{3}{4} + \frac{2}{3}\right] = \left[\frac{1}{12}, \frac{17}{12}\right] = \left[0.083, 1.417\right]. \]

- Describe How to Determine the Range for General Function Forms

For a function of the form \(\text{y}=\text{a} \cos \left( \text{b} \left( \text{x}-\text{c} \right) \right) + \text{d}\) or \(\text{y}=\text{a} \sin \left( \text{b} \left( \text{x}-\text{c} \right) \right) + \text{d}\), the range can be determined by considering the vertical shift (d) and adding and subtracting the amplitude (a) from \(d\). Thus, the range is from \( \text{d}-\text{a}\) to \( \text{d}+\text{a} \).

Key concepts

Range of Functions

To determine the range of trigonometric functions, it's key to understand the base range. The base functions, \(\text{sin}(x)\) and \(\text{cos}(x)\), range from \([-1, 1]\). Transformations like amplitude changes and vertical shifts modify this foundational range. By knowing how the function shifts and scales, we can accurately determine the range for transformed trigonometric functions.

Sine and Cosine Transformations

When dealing with sine and cosine functions, transformations can change the look and properties of the graph. These transformations could include:

• Horizontal shifts: Moving the function left or right.
• Vertical shifts: Moving the function up or down.
• Amplitude changes: Expanding or compressing the function vertically.
• Reflections: Flipping the function over a specific axis.

For example, in the function \(y=3 \cos \( x- \frac{\text{\text{\pi}}{2}} \)+5\), the cosine graph is shifted right by \(\frac{\pi}{2}\), stretched vertically by a factor of 3, and moved up by 5 units.

Amplitude and Vertical Shift

The amplitude and vertical shift are crucial for determining the range and shape of trigonometric functions. The amplitude, \(a\), represents the height from the midline to the peak or trough of the function.

To understand this, consider the general form of the functions. For \(y=a \cos \( b(x-c) \)+d\) or \(y=a \sin \( b(x-c) \)+d\):

• Amplitude (a): The coefficient before sine or cosine. It stretches the graph vertically. If \(a\) is positive, the function adheres to the typical sine or cosine wave but stretched; if negative, it's reflected.
• Vertical shift (d): The constant added or subtracted at the end. It shifts the entire function up or down by \(d\) units.

Thus, the max and min values affected by amplitude and vertical shift combine to give the function's range. It's always \( [d-a, d+a] \). For example, for \(y=1.5 \sin x +4\), the amplitude is 1.5 and the vertical shift is 4, yielding a range of \( [4-1.5, 4+1.5] = [2.5, 5.5] \).