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Chapter 5: Problem 3

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

Expert verified
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

01

- 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.
02

- 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: \[ \text{Range} = \left[ 5 - 3, 5 + 3 \right] = \left[ 2, 8 \right]. \]
03

- 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: \[ \text{Range} = \left[ -3 - 2, -3 + 2 \right] = \left[ -5, -1 \right]. \]
04

- 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: \[ \text{Range} = \left[ 4 - 1.5, 4 + 1.5 \right] = \left[ 2.5, 5.5 \right]. \]
05

- 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: \[ \text{Range} = \left[\frac{3}{4} - \frac{2}{3}, \frac{3}{4} + \frac{2}{3}\right] = \left[\frac{1}{12}, \frac{17}{12}\right] = \left[0.083, 1.417\right]. \]
06

- 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 \( \text{d}+\text{a} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 \( [4-1.5, 4+1.5] = [2.5, 5.5] \).

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Most popular questions from this chapter

Determine the phase shift and the vertical displacement with respect to \(y=\sin x\) for each function. Sketch a graph of each function. a) \(y=\sin \left(x-50^{\circ}\right)+3\) b) \(y=\sin (x+\pi)\) c) \(y=\sin \left(x+\frac{2 \pi}{3}\right)+5\) d) \(y=2 \sin \left(x+50^{\circ}\right)-10\) e) \(y=-3 \sin \left(6 x+30^{\circ}\right)-3\) f) \(y=3 \sin \frac{1}{2}\left(x-\frac{\pi}{4}\right)-10\)

A point on an industrial flywheel experiences a motion described by the function \(h(t)=13 \cos \left(\frac{2 \pi}{0.7} t\right)+15\) where \(h\) is the height, in metres, and \(t\) is the time, in minutes. a) What is the maximum height of the point? b) After how many minutes is the maximum height reached? c) What is the minimum height of the point? d) After how many minutes is the minimum height reached? e) For how long, within one cycle, is the point less than \(6 \mathrm{m}\) above the ground? f) Determine the height of the point if the wheel is allowed to turn for \(1 \mathrm{h}\) 12 min.

Sketch the graph of each function over the interval \(\left[-360^{\circ}, 360^{\circ}\right] .\) For each function, clearly label the maximum and minimum values, the \(x\) -intercepts, the \(y\) -intercept, the period, and the range. a) \(y=2 \cos x\) b) \(y=-3 \sin x\) c) \(y=\frac{1}{2} \sin x\) d) \(y=-\frac{3}{4} \cos x\)

A mass attached to the end of a long spring is bouncing up and down. As it bounces, its distance from the floor varies sinusoidally with time. When the mass is released, it takes \(0.3 \mathrm{s}\) to reach a high point of 60 cm above the floor. It takes 1.8 s for the mass to reach the first low point of \(40 \mathrm{cm}\) above the floor. a) Sketch the graph of this sinusoidal function. b) Determine the equation for the distance from the floor as a function of time. c) What is the distance from the floor when the stopwatch reads \(17.2 \mathrm{s?}\) d) What is the first positive value of time when the mass is \(59 \mathrm{cm}\) above the floor?

Without graphing, determine the amplitude and period of each function. State the period in degrees and in radians. a) \(y=2 \sin x\) b) \(y=-4 \cos 2 x\) c) \(y=\frac{5}{3} \sin \left(-\frac{2}{3} x\right)\) d) \(y=3 \cos \frac{1}{2} x\)
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