Design a sine function with the given properties.
It has a period of $12$ with a minimum value of $-4$ at $t=0$ and a maximum value of $4$ at $t=6$.

Daylight function for 40 °N Verify that the function $D(t)=2.8\(\sin\)(\(\frac{2\pi}{365}\)(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

$D\left(81\right)=12$ and $D\left(264\right)\approx 12$  (corresponding to the equinoxes).

Daylight function for 40 °N Verify that the function $D(t)=2.8\(\sin\)(\(\frac{2\pi}{365}\)(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

Its maximum and minimum values are 14.8 and 9.2, respectively, which occur approximately at $t=172$  and $t=355$, respectively (corresponding to the solstices).

Area of a circular sector Prove that the area of a sector of a circle of radius r associated with a central angle $\(\theta\)$ (measured in radians) is $A=\(\frac\)12r^2\(\theta\)$.

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{Use of Tech} Triple intersection Graph the functions f(x) = x³,g(x)=3^x, and h(x)=x^x and find their common intersection point (exactly).

Beginning with the graphs of $y=\(\sin\) x$ or $y=\(\cos\) x$, use shifting and scaling transformations to sketch the graph of the following functions. Use a graphing utility to check your work.

$q\(\left\)(x\(\right\))=3.6\(\cos\[\left\)(\(\frac{\pi x}{24}\]\right\))+2$

Daylight function for 40 °N Verify that the function $D(t)=2.8\(\sin\)(\(\frac{2\pi}{365}\)(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. It has a period of 365 days.