Chapter 7

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$y \text { -intercepts }(0, \pm 4) ; e=\frac{7}{3}$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$16(x+5)^{2}-(y-3)^{2}=1$$

Each equation defines a parabola. Without actually graphing, match the equation in Column I with its description in Column II. 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 $$y+2=-(x-4)^{2}$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$4(x+9)^{2}-25(y+6)^{2}=100$$

Each equation defines a parabola. Without actually graphing, match the equation in Column I with its description in Column II. 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 $$y=-(x-2)^{2}-4$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$9(x-2)^{2}-4(y+1)^{2}=36$$

Each equation defines a parabola. Without actually graphing, match the equation in Column I with its description in Column II. 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 $$(y-4)^{2}=x+2$$

The graph of the rational function \(y=\frac{1}{x}\) is a hyperbola that is rotated. Experiment with a graphing calculator to determine the vertices of its graph.

Each equation defines a parabola. Without actually graphing, match the equation in Column I with its description in Column II. 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 $$(y-2)^{2}=x+4$$

Find an equation for each hyperbola. \(x\) -intercepts ( \(\pm 3,0\) ); foci ( \(\pm 4,0\) )

Each equation defines a parabola. Without actually graphing, match the equation in Column I with its description in Column II. 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 $$x+2=-(y-4)^{2}$$

Find an equation for each hyperbola. \(y\) -intercepts \((0, \pm 5) ;\) foci \((0, \pm 3 \sqrt{3})\)

Each equation defines a parabola. Without actually graphing, match the equation in Column I with its description in Column II. 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 $$x=-(y-2)^{2}-4$$

Find an equation for each hyperbola. Asymptotes \(y=\pm \frac{3}{5} x ; y\) -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. $$\begin{aligned} x^{2}+y^{2} &=10 \\ 2 x^{2}-y^{2} &=17 \end{aligned}$$

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. $$\begin{aligned} x^{2}+y^{2} &=4 \\ 2 x^{2}-3 y^{2} &=-12 \end{aligned}$$

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. $$\begin{aligned} x^{2}+2 y^{2} &=9 \\ 3 x^{2}-4 y^{2} &=27 \end{aligned}$$

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. $$\begin{aligned} &2 x^{2}+3 y^{2}=5\\\ &3 x^{2}-4 y^{2}=-1 \end{aligned}$$

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. $$\begin{aligned} &2 x^{2}+2 y^{2}=20\\\ &3 x^{2}+3 y^{2}=30 \end{aligned}$$

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. $$\begin{aligned} x^{2}+y^{2} &=4 \\ 5 x^{2}+5 y^{2} &=28 \end{aligned}$$

Find an equation for each hyperbola. Foci ( \(0, \sqrt{13}\) ) and ( \(0,-\sqrt{13}\) ); 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. $$\begin{array}{l} 3 x^{2}+2 y^{2}=5 \\ x-y=-2 \end{array}$$

Find an equation for each hyperbola. Foci \((-3 \sqrt{5}, 0)\) and \((3 \sqrt{5}, 0)\); 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. $$\begin{aligned} 2 x^{2}-y^{2} &=4 \\ |x| &=|y| \end{aligned}$$

Find an equation for each hyperbola. Vertices \((4,5)\) and \((4,1)\); asymptotes \(y=\pm 7(x-4)+3\)

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

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

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. $$\begin{aligned} &2 x^{2}+3 y^{2}=5\\\ &4 x^{2}+6 y^{2}=8 \end{aligned}$$

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. $$\begin{aligned} &x^{2}+x y+y^{2}=3\\\ &x^{2}-x y+y^{2}=1 \end{aligned}$$

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

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. $$\begin{aligned} &x^{2}+2 x y-y^{2}=7\\\ &x^{2}-2 x y+y^{2}=1 \end{aligned}$$

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

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