Determine the new function after the given transformation.

$$h(x)=0.8(x-8)^2-10$$

7 units left, 4 units up

In a science class lab, a mass is suspended from a spring above a tabletop. The mass is pulled down and released. By examining frames from a video as the mass bobs up and down, the class obtains the model h(t)=5 $\sin 4(t-0.39)+10$ for the height h of the mass above the tabletop at time $t$ in seconds. Cassandra notices that after several seconds the mass is not bobbing up and down as far, but it still seems to take about the same time for a cycle. Plot $y=e^{-0.05 x}[5 \sin 4(x-0.39)]+10$ on a graphing calculator with the original model for $0 \leq x \leq 10$. State how this model differs from the original model, and how it might account for Cassandra's observation.

Explain how the domain, range, maximum values, and minimum values compare between the sine and cosine functions and their reciprocals.

Why does it make sense that a graphing calculator has a sine regression option but not a cosine regression option?

Hannah is graphing the functions f(x)=5 $\sin (x)$ and g(x)=5 $\csc (x)$ on the same coordinate plane. State what she should see if she graphs them correctly.

For the trigonometric function, determine the vertical stretch or compression and the horizontal stretch or compression. Then, plot the function and ifnd its period.

$$y=\frac{1}{3} \sin 2 x$$

Determine the new function after the given transformation.

$$q(x)=-0.5(x+2)^2-12$$

6 units down

Do transformed functions of the form $f(x)=a \tan \left(\frac{1}{b} x\right)$ have the same behavior seen in the parent function, that is, f(-x)=-f(x) ? Justify your answer and how it helps to graph transformed functions.

Graph one cycle of the transformed tangent function g(x) together with the parent function f(x)=tan x. g(x)=-$\frac{1}{4} \tan 4 x$

Determine the new function after the given transformation.

$$p(x)=0.1(x-8)^2+9$$

5 units left

Determine the period and the midline of the graph, and where the graph crosses the midline. For a sine or cosine function, find the amplitude and the maximum and minimum values and where they occur. For a tangent function, find the asymptotes and the values of the halfway points. Then plot one cycle. f(x)=3 sin $\frac{\pi}{2}(x-2)$+3

Graph one cycle of the transformed tangent function g(x) together with the parent function f(x)=tan x. g(x)=-$\tan \frac{x}{3}$

Explain why is the vertical stretch parameter, a, referred to as the amplitude when transforming sine and cosine functions, but not tangent functions. If it is not the amplitude, explain how it can be used to help graph the function.

In photosynthesis, a plant converts carbon dioxide and water to sugar and oxygen. This process is studied by measuring a plant's carbon assimilation C (in micromoles of CO$_2$ per square meter per second). For a bean plant, C(t)=1.2 $\sin \frac{\pi}{12}(t-6)+7$, where t is time in hours starting at midnight.

a. Determine the period of the function?

b. Find the maximum and the time it occurs.