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Amanda [17]
2 years ago
8

Find the period of the function. y = 5 cos one divided by twox

Mathematics
1 answer:
sesenic [268]2 years ago
4 0
\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}
\\\\
\bullet \textit{ horizontal shift by }\frac{{{  C}}}{{{  B}}}\\


\bf \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}}\\

\bf \left. \qquad  \right. if\ {{  D}}\textit{ is negative, downwards}\\\\
\left. \qquad  \right. if\ {{  D}}\textit{ is positive, upwards}\\\\
\bullet \textit{function period or frequency}\\
\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)

so.. with that template in mind, let's see

\bf \begin{array}{llll}
y=5cos&\left(\frac{1}{2}x\right)\\
&\ \uparrow \\
&\ B
\end{array}\qquad period\qquad \cfrac{2\pi }{B}\implies \cfrac{2\pi }{\frac{1}{2}}\implies \cfrac{2\pi }{1}\cdot \cfrac{2}{1}\implies 4\pi
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