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8-75.

The average high temperature for Glen Rock, PA, can be modeled using a sinusoidal function. The highest average temperature of $85^\circ\text{F}$ occurs on July 17 (Day $197$). The lowest average temperature of $37^\circ\text{F}$ occurs on January 18 (Day $17$). Write an equation that will model the temperature in Glen Rock, PA. Use the equation to estimate the high temperature on Valentines Day (Day $44$). Assume $365$ days in a year.

Plot the given points for the highest temperature and the lowest temperature.

Half of the vertical distance between the low point and the high point is the amplitude.

Identify the location of the midline. This is the vertical shift.

Double the $x$-distance between the low point and the high point for the period. Then calculate $b=\frac{2\pi}{p}$.

Using a cosine function, the horizontal shift is the horizontal distance from the $y$-axis to the first high point.

$f(x) = a\cos\left(b\left(x − h\right)\right) + k$