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Chapter 10: Problem 2

Sketch the curves of the given functions by addition of ordinates. $$y=3-2 \cos x$$

Short Answer

Expert verified
The curve of \(y = 3 - 2\cos x\) oscillates between 1 and 5, peaking at 3, with a period of \(2\pi\).

Step by step solution

01

Identify the Components

Identify the components of the function. The given function is composed of a constant term and a cosine term. Specifically, the function is expressed as \( y = 3 - 2\cos x \). This means it is a vertical translation of a scaled and reflected cosine function.
02

Sketch the Base Cosine Function

Start by sketching the basic cosine function \(y = \cos x\), which varies between -1 and 1 and crosses the y-axis at 1. It has a period of \(2\pi\), peaking at 1 and dipping to -1.
03

Apply Reflection and Scaling

Since the function inside is \(-2\cos x\), apply a reflection over the x-axis and scale the cosine function by a factor of 2. This changes the range to [-2, 2], with peaks at -2 and troughs at 2.
04

Apply Vertical Translation

Next, apply the vertical translation by adding 3 to the entire function, which adjusts the entire graph upwards by 3 units. Consequently, the peaks shift from -2 to 1, and the troughs shift from 2 to 3.
05

Determine Key Points

Identify key points on the new graph to help sketch it. At \(x = 0\), \(y = 3 - 2\cdot 1 = 1\). At \(x = \frac{\pi}{2}\), \(y = 3 - 2\cdot 0 = 3\). At \(x = \pi\), \(y = 3 - 2\cdot (-1) = 5\). Continue finding points to outline the curve.
06

Sketch the Transformed Curve

Using the key points and transformations, sketch the curve. The wave oscillates between 1 and 5 along the y-axis, with a period of \(2\pi\).

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Key Concepts

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

Cosine Function
The cosine function, denoted as \( y = \cos x \), is one of the fundamental trigonometric functions. In its most basic form, the cosine function varies between -1 and 1.
This means it reaches its maximum value of 1 when \( x = 0 \) and its minimum value of -1 at \( x = \pi \). It repeats this pattern every \( 2\pi \) units, which is known as its period.
Key characteristics of the **cosine function** include:
  • Amplitude of 1, meaning it oscillates 1 unit above and below the x-axis.
  • Period of \( 2\pi \), indicating the length of each complete wave.
  • The axis of symmetry along the y-axis, causing it to have an even symmetry.
Understanding this basic shape and motion helps as we look into transforming it through reflection, scaling, and translation.
Vertical Translation
Vertical translation involves shifting the entire graph of a function up or down along the y-axis. In the given function, \( y = 3 - 2\cos x \), the term '+3' causes an upward shift.
This means that every point on the graph moves up by 3 units.
Vertical translation does not change the shape of the graph but moves it uniformly:
  • The peaks and troughs of the wave shift upward, maintaining their spacing.
  • For the function \( y = -2\cos x + 3 \), peaks initially at -2 now appear at 1 (from -2 + 3 = 1).
  • Troughs which were at 2 now shift to 5 (from 2 + 3 = 5).
Vertical translations are crucial when altering cosine functions as they affect the range, moving the waveform vertically within the coordinate plane.
Reflection and Scaling
Reflection and scaling are transformations that modify the appearance and dimension of a trigonometric graph.
Specifically, for the function \( y = 3 - 2\cos x \), these transformations drastically change how the base cosine waveform is displayed.
**Reflection** is the flipping of the graph over an axis:
  • By multiplying \( \cos x \) by -2, the graph reflects over the x-axis, flipping the peaks and troughs.
**Scaling** affects the amplitude:
  • The scaling of -2 increases the amplitude from 1 to 2, meaning the peaks and troughs now reach twice as far from the axis.
In combination, reflection and scaling create a broader and inverted cosine wave that underlies the resulting function.
Graph Sketching
Sketching a transformed cosine function involves understanding and applying the changes due to transformations.
For the function \( y = 3 - 2\cos x \), we take a step-by-step approach:
  • Start with the base: Know that the base cosine function \( \cos x \) oscillates between -1 and 1.
  • Apply reflection and scaling: Reflect about the x-axis and scale by multiplying by -2, resulting in \( -2\cos x \) with a range of [-2, 2].
  • Translate vertically: Add 3 to shift the entire graph upward, now oscillating between 1 and 5.
  • Plot Key Points: Identify and plot points at key angles like 0, \(\frac{\pi}{2}\), and \(\pi\) to nail down the shape.
  • Complete the curve: Use these transformations and points to outline the new waveform across its period of \(2\pi\).
Successfully sketching such a curve involves understanding how each transformation affects the shape and position of the original cosine function.

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Most popular questions from this chapter

Sketch the appropriate graphs, and check each on a calculator. Near Antarctica, an iceberg with a vertical face \(200 \mathrm{m}\) high is seen from a small boat. At a distance \(x\) from the iceberg, the angle of elevation \(\theta\) of the top of the iceberg can be found from the equation \(x=200 \cot \theta .\) Sketch \(x\) as a function of \(\theta\).

Solve the given problems. In Exercises 41 and 42 use a calculator to view the indicated curves. In performing a test on a patient, a medical technician used an ultrasonic signal given by the equation \(I=A \sin (\omega t+\theta) .\) View two cycles of the graph of \(I\) vs. \(t\) if \(A=5 \mathrm{n} \mathrm{W} / \mathrm{m}^{2}, \omega=2 \times 10^{5} \mathrm{rad} / \mathrm{s}\) and \(\theta=0.4\)

View at least two cycles of the graphs of the given functions on a calculator. $$y=-2 \cot \left(2 x+\frac{\pi}{6}\right)$$

Solve the given problems. To tune the instruments of an orchestra before a concert, an A note is struck on a piano. The piano wire vibrates with a displacement \(y\) (in \(\mathrm{mm}\) ) given by \(y=3.20 \cos 880 \pi t,\) where \(t\) is in seconds. Sketch the graph of \(y\) vs. \(t\) for \(0 \leq t \leq 0.01 \mathrm{s}\)

Sketch the required curves. The sinusoidal electromagnetic wave emitted by an antenna in a cellular phone system has a frequency of \(7.5 \times 10^{9} \mathrm{Hz}\) and an amplitude of \(0.045 \mathrm{V} / \mathrm{m} .\) Find the equation representing the wave if it starts at the origin. Sketch two cycles.
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