Write parametric equations for the given curve for the given definitions of $x$:

$$y=-4 x+1$$

a. $x=t$ b. $x=\frac{t}{2}$ c. $x=-4 t$

Write parametric equations for the given curve for the given definitions of $x$:

$$y=2 x-4$$

a. $x=t$ b. $x=\frac{t}{3}$ c. $x=-3t$

The given equation represents a conic section (nondegenerative case). Identify the type of conic section: a. $4 x^2-12 x y+6 y^2+2 x-3 y-8=0$ b. $4 x^2-12 x y+9 y^2+2 x-3 y-8=0$

a. Identify the center of the ellipse. b. Identify the vertices. c. Identify the endpoints of the minor axis. d. Identify the foci. e. Graph the ellipse.

$$\frac{4(x-3)^{2}}{25}+\frac{16(y-2)^{2}}{49}=1$$

a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola.

$$x^{2}-\frac{(y-4)^{2}}{8}=1$$

Describe the graph of the equation. $x=-4 \sqrt{1-\frac{y^{2}}{9}}$

An equation of a parabola $x^{2}=4 p y \text { or } y^{2}=4 p x$ is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. $4 x^{2}=40 y$

Eliminate the parameter and write an equation in rectangular coordinates to represent the given curve: Hyperbola: $x=h+a \sec \theta$ and $y=k+b \tan \theta$

a. Identify the center. b. Identify the vertices. c. Identify the foci. d. Write equations for the asymptotes. e. Graph the hyperbola.

$$y^{2}-\frac{(x-2)^{2}}{18}=1$$

An equation of a parabola $x^{2}=4 p y \text { or } y^{2}=4 p x$ is given. a. Identify the vertex, value of p, focus, and focal diameter of the parabola. b. Identify the endpoints of the latus rectum. c. Graph the parabola. d. Write equations for the directrix and axis of symmetry. $3 y^{2}=12 x$

Write an equation of the conic rotated about the origin through the given angle. Then graph the result on a graphing utility.

Given: $r=\frac{5}{2+2 \sin \theta}$; Counterclockwise $\frac{4 \pi}{3}$

An equation of a paraboLa is given in standard form. a. Determine the value of p. b. Identify the vertex. c. Identify the focus. d. Determine the focal diameter. e. Determine the endpoints of the latus rectum. f. Write an equation of the directrix. g. Write an equation for the axis of symmetry. h. Graph the parabola. $x^{2}=-2 y$

Solve the system of equations.

$$\begin{aligned} &3 x^{2}-4 y^{2}=-13\\ &5 x^{2}+2 y^{2}=13 \end{aligned}$$

Use Cramer's rule lo solve for z.

$$\begin{aligned} 3 x-4 y+z &=3 \\ 2 x+3 y & \quad \quad =8 \\ 5 y+8 z &=11 \end{aligned}$$