Chapter 8

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$x=-(y-2)^{2}-4$$ A. Vertex \((2,-4) ;\) opens downward B. Vertex \((2,-4) ;\) opens upward C. Vertex \((4,-2)\); opens downward D. Vertex \((4,-2)\); opens upward E. Vertex \((-2,4)\); opens left F. Vertex \((-2,4)\); opens right G. Vertex \((-4,2) ;\) opens left H. Vertex \((-4,2)\); opens right

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}+y^{2} &=10 \\ 2 x^{2}-y^{2} &=17 \end{aligned}$$

Find an equation for each hyperbola. $$\text { Asymptotes } y=\pm \frac{3}{5} x, y \text { -intercepts }(0, \pm 3)$$

For the graph of \((x-h)^{2}=4 c(y-k)\) in what quadrant is the vertex for each condition? (a) \(h<0, k<0\) (b) \(h<0, k>0\) (c) \(h>0, k<0\) (d) \(h>0, k>0\)

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}+y^{2} &=4 \\ 2 x^{2}-3 y^{2} &=-12 \end{aligned}$$

Find an equation for each hyperbola. y-intercept \((0,-2) ;\) center at origin; passing through \((2,3)\)

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}+2 y^{2} &=9 \\ 3 x^{2}-4 y^{2} &=27 \end{aligned}$$

Find an equation for each hyperbola. Vertices \((0,6)\) and \((0,-6) ;\) asymptotes \(y=\pm \frac{1}{2} x\)

Give the focus, directrix, and axis of each parabola. $$x^{2}=16 y$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &2 x^{2}+3 y^{2}=5\\\ &3 x^{2}-4 y^{2}=-1 \end{aligned}$$

Find an equation for each hyperbola. Vertices \((-10,0)\) and \((10,0) ;\) asymptotes \(y=\pm 5 x\)

Give the focus, directrix, and axis of each parabola. $$x^{2}=4 y$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &2 x^{2}+2 y^{2}=20\\\ &3 x^{2}+3 y^{2}=30 \end{aligned}$$

Find an equation for each hyperbola. Vertices \((-3,0)\) and \((3,0) ;\) passing through \((6,1)\)

Give the focus, directrix, and axis of each parabola. $$x^{2}=-\frac{1}{2} y$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}+y^{2} &=4 \\ 5 x^{2}+5 y^{2} &=28 \end{aligned}$$

Find an equation for each hyperbola. Vertices \((0,5)\) and \((0,-5) ;\) passing through \((3,10)\)

Give the focus, directrix, and axis of each parabola. $$x^{2}=\frac{1}{9} y$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} 3 x^{2}+2 y^{2} &=5 \\ x-y &=-2 \end{aligned}$$

Find an equation for each hyperbola. $$\text { Foci }(0, \sqrt{13}) \text { and }(0,-\sqrt{13}) ; \text { asymptotes } y=\pm 5 x$$

Give the focus, directrix, and axis of each parabola. $$y^{2}=\frac{1}{16} x$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} 2 x^{2}-y^{2} &=4 \\ |x| &=|y| \end{aligned}$$

Find an equation for each hyperbola. $$\text { Foci }(-3 \sqrt{5}, 0) \text { and }(3 \sqrt{5}, 0) ; \text { asymptotes } y=\pm 2 x$$

Give the focus, directrix, and axis of each parabola. $$y^{2}=-\frac{1}{32} x$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &x^{2}+y^{2}=8\\\ &x^{2}-y^{2}=0 \end{aligned}$$

Give the focus, directrix, and axis of each parabola. $$y^{2}=-16 x$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &2 x^{2}+3 y^{2}=5\\\ &4 x^{2}+6 y^{2}=8 \end{aligned}$$

Find an equation for each hyperbola. Vertices \((5,-2)\) and \((1,-2) ;\) asymptotes \(y=\pm \frac{3}{2}(x-3)-2\)

Give the focus, directrix, and axis of each parabola. $$y^{2}=-4 x$$

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &x^{2}+x y+y^{2}=3\\\ &x^{2}-x y+y^{2}=1 \end{aligned}$$

Find an equation for each hyperbola. Center \((1,-2) ;\) focus \((4,-2) ;\) vertex \((3,-2)\)

Write an equation for each parabola with vertex at the origin. Focus \((0,-2)\)

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &x^{2}+2 x y-y^{2}=7\\\ &x^{2}-2 x y+y^{2}=1 \end{aligned}$$

Find an equation for each hyperbola. Center \((9,-7) ;\) focus \((9,3) ;\) vertex \((9,-1)\)

Write an equation for each parabola with vertex at the origin. Focus \((5,0)\)

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} x^{2}-x y+y^{2} &=5 \\ 2 x^{2}+x y-y^{2} &=10 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$x^{2}-2 x-y^{2}+2 y=4$$

Write an equation for each parabola with vertex at the origin. Focus \(\left(-\frac{1}{2}, 0\right)\)

Solve each nonlinear system of equations analytically for all real solutions. $$\begin{aligned} &x^{2}+3 x y+y^{2}=5\\\ &x^{2}-2 x y-y^{2}=-7 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$y^{2}+4 y-x^{2}+2 x=6$$

Write an equation for each parabola with vertex at the origin. Focus \(\left(0, \frac{1}{4}\right)\)

Use substitution to solve the nonlinear system of equations in three variables. Note that solutions are ondered triples. $$\begin{aligned} 2 x^{2}+y^{2}+3 z^{2} &=3 \\ 2 x+y-z &=1 \\ x+y &=0 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$3 y^{2}+24 y-2 x^{2}+12 x+24=0$$

Write an equation for each parabola with vertex at the origin. Through \((2,-2 \sqrt{2}) ;\) opening to the right

Use substitution to solve the nonlinear system of equations in three variables. Note that solutions are ondered triples. $$\begin{aligned} &x^{2}+y^{2}+z^{2}=4\\\ &x+y+z=2\\\ &x-y \quad=0 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$4 x^{2}+16 x-9 y^{2}+18 y=29$$

Write an equation for each parabola with vertex at the origin. Through \((\sqrt{3}, 3) ;\) opening upward

Use substitution to solve the nonlinear system of equations in three variables. Note that solutions are ondered triples. $$\begin{aligned} &\frac{x^{2}}{16}+\frac{y^{2}}{4} \leq 1\\\ &x^{2}-y^{2} \geq 1 \end{aligned}$$

Write the equation in standard form for a hyperbola centered at ( \(h, k\) ). Identify the center and vertices. $$x^{2}-6 x-2 y^{2}+7=0$$

Write an equation for each parabola with vertex at the origin. Through \((\sqrt{10},-5) ;\) opening downward