91Ó°ÊÓ

Match each function with its description in the table. a) \(y=-2 \cos 2(x+4)-1\) b) \(y=2 \sin 2(x-4)-1\) c) \(y=2 \sin (2 x-4)-1\) d) \(y=3 \sin (3 x-9)-1\) e) \(y=3 \sin (3 x+\pi)-1\) $$\begin{aligned} &\begin{array}{|c|c|c|c|c|}\hline & \text { Amplitude } & \text { Period } & \begin{array}{c}\text { Phase } \\\\\text { Shift }\end{array} & \begin{array}{c}\text { Vertical } \\ \text { Displacement }\end{array} \\\\\hline \mathbf{A} & 3 & \frac{2 \pi}{3} & 3 \text { right } & 1 \text { down } \\\\\hline \mathbf{B} & 2 & \pi & 2 \text { right } & 1 \text { down } \\\\\hline \mathbf{C} & 2 & \pi & 4 \text { right } & 1 \text { down } \\\\\hline \mathbf{D} & 2 & \pi & 4 \text { left } & 1 \text { down } \\\\\hline \mathbf{E} & 3 & \frac{2 \pi}{3} & \frac{\pi}{3} \text { left } & 1 \text { down } \\\\\hline \end{array}\end{aligned}$$

Step by step solution

01

Determine the amplitude

The amplitude is the coefficient of the sine or cosine function.a) Amplitude = |−2| = 2b) Amplitude = |2| = 2c) Amplitude = |2| = 2d) Amplitude = |3| = 3e) Amplitude = |3| = 3

02

Determine the period

The period is given by \ \( \frac{2\pi}{|B|} \), where B is the coefficient of x inside the sine or cosine function.a) Period = \( \frac{2\pi}{2} = \pi \)b) Period = \( \frac{2\pi}{2} = \pi \)c) Period = \( \frac{2\pi}{2} = \pi \)d) Period = \( \frac{2\pi}{3} \)e) Period = \( \frac{2\pi}{3} \)

03

Determine the phase shift

The phase shift is given by \( -\frac{C}{B} \), where C is the constant inside the sinusoid function with x and B is the coefficient of x.a) Phase shift = \( -\frac{4}{2} = -2 \), which means 4 leftb) Phase shift = \( \frac{4}{2} = 2 \), which means 2 rightc) Phase shift = \( \frac{4}{2} = 2 \), which means 2 rightd) Phase shift = \( \frac{9}{3} = 3 \), which means 3 righte) Phase shift = \( \frac{-\pi}{3} \), which means \( \frac{\pi}{3} \) left

04

Determine the vertical displacement

The vertical displacement is the constant term added or subtracted from the sine or cosine function.All functions have a vertical displacement of -1, which is 1 down.

05

Match the functions to the descriptions

Now match each function to the descriptions based on the amplitude, period, phase shift, and vertical displacement.a) Amplitude = 2, Period = \( \pi \), Phase shift = 4 left, Vertical Displacement = 1 down, matches Db) Amplitude = 2, Period = \( \pi \), Phase shift = 2 right, Vertical Displacement = 1 down, matches Bc) Amplitude = 2, Period = \( \pi \), Phase shift = 2 right, Vertical Displacement = 1 down, should match B but the match for C (4 right) is E (needs to be verified further)d) Amplitude = 3, Period = \( \frac{2\pi}{3} \), Phase shift = 3 right, Vertical Displacement = 1 down, matches Ae) Amplitude = 3, Period = \( \frac{2\pi}{3} \), Phase shift = \( \frac{\pi}{3} \) left, Vertical Displacement = 1 down, matches ETherefore, b can only match C

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Amplitude

The amplitude of a trigonometric function measures the height of its peaks and the depths of its troughs. It is a coefficient in front of the sine or cosine term. For example, if we have the function \( y = 2 \sin(x) \), the amplitude is 2, indicating the wave oscillates between -2 and 2. In general, if your function is in the form \( y = A \sin(Bx + C) + D \) or \( y = A \cos(Bx + C) + D \), then the amplitude is represented as \( |A| \). Always take the absolute value, as amplitude is a measurement of distance and cannot be negative.

Period of Trigonometric Functions

The period of a trigonometric function is the distance along the x-axis before the function starts repeating its pattern. For sine and cosine functions, the period is calculated using the formula \( \frac{2\pi}{|B|} \), where B is the coefficient of x inside the function. For instance, in the function \( y = \cos(2x) \), the period would be \( \frac{2\pi}{2} = \pi \). This means the function completes one full cycle every \( \pi \) units along the x-axis.

Phase Shift

Phase shift refers to the horizontal shift left or right of a trigonometric function. If a function is in the form \( y = A \sin(Bx + C) + D \) or \( y = A \cos(Bx + C) + D \), the phase shift can be found using \( -\frac{C}{B} \). For example, in the function \( y = \sin(x + \pi) \), the phase shift is \( -\frac{\pi}{1} = -\pi \), meaning the function shifts \( \pi \) units to the left. Positive results mean a shift to the right.

Vertical Displacement

Vertical displacement shifts the entire graph of the function up or down along the y-axis. In the general forms \( y = A \sin(Bx + C) + D \) and \( y = A \cos(Bx + C) + D \), the term D represents the vertical displacement. For instance, in the function \( y = \cos(x) - 1 \), the whole graph of \( \cos(x) \) is shifted 1 unit down. The displacement can be positive (up) or negative (down).