Question

Determine the amplitude, period, and phase shift.
y=1/2cos(3x+pi/2)

Answer 1

well... I should say \(\qquad \qquad \qquad \qquad \textit{function transformations}
\\ \quad \\
% function transformations for trigonometric functions
\begin{array}{rllll}
% left side templates
f(x)=&{\color{purple}{ A}}cos({\color{blue}{ B}}x+{\color{red}{ C}})+{\color{green}{ D}}
\\ \quad \\

\end{array}\qquad

\begin{array}{llll}
% right side info
\bullet \textit{ stretches or shrinks horizontally by amplitude } |{\color{purple}{ A}}|\\\
\bullet \textit{ horizontal shift by }\frac{{\color{red}{ C}}}{{\color{blue}{ B}}}\\
\qquad if\ \frac{{\color{red}{ C}}}{{\color{blue}{ B}}}\textit{ is negative, to the right}\\
\qquad if\ \frac{{\color{red}{ C}}}{{\color{blue}{ B}}}\textit{ is positive, to the left}\\
\bullet \textit{vertical shift by }{\color{green}{ D}}\\
\qquad if\ {\color{green}{ D}}\textit{ is negative, downwards}\\
\qquad if\ {\color{green}{ D}}\textit{ is positive, upwards}\\
\bullet \textit{function period}\\
\qquad \frac{2\pi }{{\color{blue}{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\
\qquad \frac{\pi }{{\color{blue}{ B}}}\ for\ tan(\theta),\ cot(\theta)
\end{array}\)

what do you get from that then?

Answer 2

hmmm anyway...
\(\qquad \qquad \qquad \qquad \textit{function transformations}
\\ \quad \\
% function transformations for trigonometric functions
\begin{array}{rllll}
% left side templates
f(x)=&{\color{purple}{ \frac{1}{2}}}cos({\color{blue}{ 3}}x+{\color{red}{ \frac{\pi }{2}}})+{\color{green}{ 0}}
\\ \quad \\

\end{array}\qquad

\begin{array}{llll}
% right side info
\bullet \textit{ stretches or shrinks}\\\quad \textit{ horizontally by amplitude } |{\color{purple}{ A}}|\\\
\bullet \textit{ horizontal shift by }\frac{{\color{red}{ C}}}{{\color{blue}{ B}}}\\
\qquad if\ \frac{{\color{red}{ C}}}{{\color{blue}{ B}}}\textit{ is negative, to the right}\\
\qquad if\ \frac{{\color{red}{ C}}}{{\color{blue}{ B}}}\textit{ is positive, to the left}\\
\bullet \textit{vertical shift by }{\color{green}{ D}}\\
\qquad if\ {\color{green}{ D}}\textit{ is negative, downwards}\\
\qquad if\ {\color{green}{ D}}\textit{ is positive, upwards}\\
\bullet \textit{function period}\\
\qquad \frac{2\pi }{{\color{blue}{ B}}}\ for\ cos(\theta),\ sin(\theta),\ sec(\theta),\ csc(\theta)\\
\qquad \frac{\pi }{{\color{blue}{ B}}}\ for\ tan(\theta),\ cot(\theta)
\end{array}\)

see what you get