Question

A sound wave is modeled with the equation y = 1/4 cos 2pi/3 theta .
A. Find the period. Explain your method.
B. Find the amplitude. Explain your method.
C. What is the equation of the midline? What does it represent?

Answer 1

$\bf \qquad \qquad \qquad \qquad \textit{function transformations} \\ \quad \\ % function transformations for trigonometric functions \begin{array}{rllll} % left side templates f(x)=&{{ A}}sin({{ B}}x+{{ C}})+{{ D}} \\\\ f(x)=&{{ A}}cos({{ B}}x+{{ C}})+{{ D}}\\\\ f(x)=&{{ A}}tan({{ B}}x+{{ C}})+{{ D}} \end{array} \\\\ -------------------$

$\bf \bullet \textit{ stretches or shrinks}\\ \left. \qquad \right. \textit{horizontally by amplitude } |{{ A}}|\\\\ \bullet \textit{ flips it upside-down if }{{ A}}\textit{ is negative}\\ \left. \qquad \right. \textit{reflection over the x-axis} \\\\ \bullet \textit{ flips it sideways if }{{ B}}\textit{ is negative}\\ \left. \qquad \right. \textit{reflection over the y-axis}$

$\bf \bullet \textit{ horizontal shift by }\frac{{{ C}}}{{{ B}}}\\ \left. \qquad \right. if\ \frac{{{ C}}}{{{ B}}}\textit{ is negative, to the right}\\\\ \left. \qquad \right. if\ \frac{{{ C}}}{{{ B}}}\textit{ is positive, to the left}\\\\ \bullet \textit{vertical shift by }{{ D}}\\ \left. \qquad \right. if\ {{ D}}\textit{ is negative, downwards}\\\\ \left. \qquad \right. if\ {{ D}}\textit{ is positive, upwards}\\\\ \bullet \textit{function period or frequency}$

$\bf \left. \qquad \right. \frac{2\pi }{{{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\\\ \left. \qquad \right. \frac{\pi }{{{ B}}}\ for\ tan(\theta),\ cot(\theta)$

now, with that template in mind, let's see

$\bf \begin{array}{llll} y=&\frac{1}{4}cos&\left( \frac{2\pi }{3}\theta \right)\\ &\uparrow &\quad \uparrow\\ &A&\quad B \end{array} \\\\ -------------------------------\\\\ A)\qquad period=\cfrac{2\pi }{B}\implies \cfrac{2\pi }{\frac{2\pi }{3}}\implies \cfrac{\frac{2\pi}{1} }{\frac{2\pi }{3}}\implies \cfrac{2\pi }{1}\cdot \cfrac{3}{2\pi }$

and surely you know how much that is

for part B)... well, is right there, from the template what A is

for part C)

well, the equation has no vertical shifting, so the midline is the same as for the parent function cos(θ).

Answer 2

Answer:

A) Period = 3

B) Amplitude = $\frac{1}{4}$

C) The equation of midline y=0

Step-by-step explanation:

Given : A sound wave is modeled with the equation $y =\frac{1}{4}cos (\frac{2\pi}{3}) \theta$

To find :

A) Period

B)Amplitude

C) The equation of Midline

Solution :

The general formula for cosine is:  

$y=Acos(Bx)+C$

Where A is Amplitude

$B=\frac{2\pi}{\text{Period}}$

C is Mid line

Comparing the given function with general form of cosine we get,

$y =\frac{1}{4}cos (\frac{2\pi}{3}) \theta$

A) Period - $B=\frac{2\pi}{3}$

and we know, $B=\frac{2\pi}{\text{Period}}$

$\frac{2\pi}{3}=\frac{2\pi}{\text{Period}}$

$\text{Period}=\frac{2\pi\times 3}{2\pi}$

$\text{Period}=3$

B) Amplitude- $A=\frac{1}{4}$

C) The equation of midline

Midline is C=0

The equation of midline is y=0.