Graph the function over a one-period interval.
-Suppose that the average monthly low temperatures for a small town are shown in the table.
$\begin{array}{l|llllllllllll}
\text { Month } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \
\hline \text { Temperature }\left({ }^{\circ} \mathrm{F}\right) & 19 & 27 & 38 & 45 & 57 & 62 & 65 & 58 & 51 & 41 & 33 & 25
\end{array}$

Model this data using $f ( x ) = a \sin ( b ( x - c ) ) + d$.
A) $f ( x ) = 42 \sin \left( \frac { \pi } { 6 } ( x - 4 ) \right) + 23$
B) $f ( x ) = 23 \sin \left( \frac { \pi } { 12 } ( x - 4 ) \right) + 42$
C) $f ( x ) = 23 \sin \left( \frac { \pi } { 6 } ( x - 7 ) \right) + 42$
D) $f ( x ) = 23 \sin \left( \frac { \pi } { 6 } ( x - 4 ) \right) + 42$

Question

Graph the function over a one-period interval.
-Suppose that the average monthly low temperatures for a small town are shown in the table.
$\begin{array}{l|llllllllllll}
\text { Month } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \
\hline \text { Temperature }\left({ }^{\circ} \mathrm{F}\right) & 19 & 27 & 38 & 45 & 57 & 62 & 65 & 58 & 51 & 41 & 33 & 25
\end{array}$

Model this data using $f ( x ) = a \sin ( b ( x - c ) ) + d$.
A) $f ( x ) = 42 \sin \left( \frac { \pi } { 6 } ( x - 4 ) \right) + 23$
B) $f ( x ) = 23 \sin \left( \frac { \pi } { 12 } ( x - 4 ) \right) + 42$
C) $f ( x ) = 23 \sin \left( \frac { \pi } { 6 } ( x - 7 ) \right) + 42$
D) $f ( x ) = 23 \sin \left( \frac { \pi } { 6 } ( x - 4 ) \right) + 42$

Quizplus · Accepted Answer

The correct model for the data is $f(x) = 23 \sin\left(\frac{\pi}{6}(x - 4)\right) + 42$. This is because the amplitude of the temperature variation is 23 (half the difference between the maximum and minimum temperatures), the period is 12 months (which corresponds to $\frac{2\pi}{b} = 12$, solving for $b$ gives $\frac{\pi}{6}$), the phase shift is to the right by 4 months to align with the peak in July (month 7), and the vertical shift is 42, which is the average value of the maximum and minimum temperatures.

Graph the Function Over a One-Period Interval Model This Data Using