$$\text { Find an isomorphism from } Z _ { 12 } \text { to } Z _ { 4 } \oplus Z _ { 3 }$$

Verify each identity.

$$\tan \frac { \alpha } { 2 } = \frac { \tan \alpha } { \sec \alpha + 1 }$$

Find the vertex of each parabola. focus: (-2, -1); directrix: y = -6

Use a graphing utility to graph the trigonometric function. Use the graph to make a conjecture about a simplification of the expression. Verify the resulting identity algebraically. $y=\frac{1}{\cot x+1}+\frac{1}{\tan x+1}$

use an inverse matrix to solve (if possible) the system of linear equations. {4x - 2y = -10 , -19x + 9y = 47

Find and classify the stationary points on the curve

$$y = \sin x + 4 \cos x$$

in the interval

$$0 < x < 2 \pi$$

.

Use Cramer's rule to solve each system.

$$\left\{ \begin{array} { r r } { 2 x + 2 y + 3 z = 10 } \\ { 4 x - y + z = - 5 } \\ { 5 x - 2 y + 6 z = 1 } \end{array} \right.$$

Choose one type of quick bread and write a one-page essay about how to bake that type of quick bread.

Derrick is designing triangular panes for a stained glass window. Determine the number of different triangles that he can form using the given measurements. Then solve the triangles. Round to the nearest tenth.

$$a = 3 \mathrm { cm } , b = 4.6 \mathrm { cm } , \mathrm { m } \angle A = 95 ^ { \circ }$$

Explain what transformations you would need to apply to the graph of

$$y=f\left(x\right)$$

to graph each function. a)

$$y=3f\left(x\right)-1$$

b)

$$y=f\left(x-2\right)+3$$

c)

$$y=f\left(2x\right)-5$$

d)

$$y=-f\left(\frac{1}{2}x\right)-2$$

e)

$$y=\frac{2}{3}f\left(x+3\right)+1$$

f)

$$y=4f\left(-x\right)-4$$

What mass of

$$\mathrm { H } _ { 2 } \mathrm { SO } _ { 4 }$$

is contained in 60.00 mL of a 5.85 M solution of sulfuric acid?

Find

$$\sin \frac { x } { 2 } ,\cos \frac { x } { 2 }$$

, and

$$\tan \frac { x } { 2 }$$

from the given information.

$$\cos x = - \frac { 4 } { 5 } , \quad 180 ^ { \circ } < x < 270 ^ { \circ }$$

Simplify: 3 5/6 - 1/3

Find each volume. Round to the nearest tenth, if necessary. l = 8$\frac{1}{2}$ m, w = 30 m, h = 15 m