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.

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

State the key features of the graphs of a trigonometric function, and how you can use a trigonometric function to model real-world data.

Consider the radian measure $\frac{23 \pi}{4}$. Determine whether each statement is correct.

A. $\cos \frac{23 \pi}{4} \approx$-0.707

B. $\tan \frac{23 \pi}{4}$=-1

C. $\sin \frac{23 \pi}{4} \approx$ 0.707

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

$$y=-2 \cos \frac{1}{3} x$$

State a rotation that will map the graph of $f(\theta)=\sin \theta$ onto itself on the interval $0 \leq \theta \leq 2 \pi$.

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

For the situation described: (a) Plot the data, roughly fit a sine model of the form $y=a \sin \frac{1}{b}(x-h)+k$ to the data, and plot the model on the same grid as the data.

(b) Determine a sine regression model using a graphing calculator, and then visually compare the two models in relation to the data.

$(c)$ Use a graphing calculator and the regression model to answer the question.

The table shows the decimal portion of the moon's surface illuminated from Earth's view at midnight from Washington, D.C., in March, 2014, where x represents the day of the month.

A "gibbous moon" refers to the moon phases when more than half of the moon is illuminated, but it is not a full moon. Determine about how long is the period that the moon is either a gibbous moon or a full moon.

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.

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

$$y=-3 \sin \frac{1}{6} x$$

Determine the minimum and maximum values of $f(\theta)=\sin \theta$ and $f(\theta)=\cos \theta$ on the interval $0 \leq \theta \leq 2 \pi$ . Determine where the extreme values occur in relation to the $\theta$-intercepts.

For the function $f(x)=2 \cos 2\left(x-\frac{\pi}{2}\right)+1$, determine the period, the midline, where the graph crosses the midline, the amplitude, and the maximum and minimum values and where they occur. Then sketch one cycle of the function.

Using reference angles determine why the graph of $f(\theta)=\tan \theta$ appears to show $f(-\theta)=-\tan \theta$.

Determine the new function for $f(x)=(x+1)^2-2$ after the given transformation. vertical stretch, factor of 2 factor of 10.