gaokao 2015 Q18

gaokao · China · hubei-arts Trig Graphs & Exact Values
18. A student uses the ``five-point method'' to draw the graph of the function $f ( \mathrm { x } ) = \mathrm { A } \sin ( \omega \mathrm { x } + \varphi ) \left( \omega > 0 , \varphi < \frac { \pi } { 2 } \right)$ during a certain period, creates a table and fills in partial data as follows:
$\omega \mathrm { x } + \varphi$0$\frac { \pi } { 2 }$$\pi$$\frac { 3 \pi } { 2 }$$2 \pi$
x$\frac { \pi } { 3 }$$\frac { 5 \pi } { 6 }$
$\mathrm {~A} \sin ( \omega \mathrm { x } + \varphi )$05- 50

(I) Please complete the above data, fill in the corresponding positions on the answer sheet, and directly write the analytical expression of the function $f ( \mathrm { x } )$; (II) Shift all points on the graph of $y = f ( \mathrm { x } )$ to the left by $\frac { \pi } { 6 }$ units to obtain the graph of $y = g ( \mathrm { x } )$. Find the center of symmetry of the graph of $y = g ( \mathrm { x } )$ that is closest to the origin O.
18. A student uses the ``five-point method'' to draw the graph of the function $f ( \mathrm { x } ) = \mathrm { A } \sin ( \omega \mathrm { x } + \varphi ) \left( \omega > 0 , \varphi < \frac { \pi } { 2 } \right)$ during a certain period, creates a table and fills in partial data as follows:

\begin{center}
\begin{tabular}{ | c | l | l | l | l | l | }
\hline
$\omega \mathrm { x } + \varphi$ & 0 & $\frac { \pi } { 2 }$ & $\pi$ & $\frac { 3 \pi } { 2 }$ & $2 \pi$ \\
\hline
x &  & $\frac { \pi } { 3 }$ &  & $\frac { 5 \pi } { 6 }$ &  \\
\hline
$\mathrm {~A} \sin ( \omega \mathrm { x } + \varphi )$ & 0 & 5 &  & - 5 & 0 \\
\hline
\end{tabular}
\end{center}

(I) Please complete the above data, fill in the corresponding positions on the answer sheet, and directly write the analytical expression of the function $f ( \mathrm { x } )$;\\
(II) Shift all points on the graph of $y = f ( \mathrm { x } )$ to the left by $\frac { \pi } { 6 }$ units to obtain the graph of $y = g ( \mathrm { x } )$. Find the center of symmetry of the graph of $y = g ( \mathrm { x } )$ that is closest to the origin O.\\