Chapter 7

Give two parametric representations for each plane curve. Use your calculator to verify your results. $$x=\sqrt{y+1}$$

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$x^{2}+y^{2}=81$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$12 x^{2}+9 y^{2}=36$$

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$x^{2}-6 x+9 y^{2}=0$$

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$x^{2}+(y+3)^{2}=49$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$8 x^{2}-y^{2}=16$$

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$4 x^{2}+16 x+5 y^{2}-10 y+1=0$$

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$(x-3)^{2}+(y-2)^{2}=25$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$x^{2}-y^{2}=4$$

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$2 x^{2}+4 x+3 y^{2}-18 y+23=0$$

AProjectile If a projectile is fired at an angle of \(30^{\circ}\) with the horizontal, the parametric equations that describe its motion are $$x=v_{0} \frac{\sqrt{3}}{2} t, y=\frac{v_{0}}{2} t-16 t^{2}, \quad \text { for } t \text { in }[0, \infty)$$

Graph each circle using a graphing calculator. Use a square viewing window. Give the domain and range. $$(x+2)^{2}+(y+3)^{2}=36$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$x^{2}+2 y^{2}=8$$

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$16 x^{2}-16 x+4 y^{2}+12 y=51$$

Path of a Projectile \(\quad\) A projectile moves so that its position at any time \(t\) is given by the equations $$x=60 t \text { and } y=80 t-16 t^{2}$$ Graph the path of the projectile, and find the equivalent rectangular equation. Use the window \([0,300]\) by \([0,200]\)

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}+6 x+y^{2}+8 y+9=0$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$4 x^{2}+7 y^{2}=28$$

Write the equation in standard form for an ellipse centered at ( \(h, k\) ). Identify the center and vertices. $$16 x^{2}+48 x+4 y^{2}-20 y+57=0$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}+8 x+y^{2}-6 y+16=0$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$9 x^{2}-y^{2}=1$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}-4 x+y^{2}+12 y=-4$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$x^{2}-9 y^{2}=18$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{y^{2}}{9}-\frac{x^{2}}{9}=1$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}-12 x+y^{2}+10 y=-25$$

Find the eccentricity \(e\) of each ellipse or hyperbola. $$x^{2}+10 y^{2}=10$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$49 y^{2}-36 x^{2}=1764$$

Give two parametric representations of the line through the point \(\left(x_{1}, y_{1}\right)\) with slope \(m\)

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$4 x^{2}+4 x+4 y^{2}-16 y-19=0$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$144 x^{2}-49 y^{2}=7056$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$9 x^{2}+12 x+9 y^{2}-18 y-23=0$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{4 x^{2}}{9}-\frac{25 y^{2}}{16}=1$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}+2 x+y^{2}-6 y+14=0$$

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$\text { Focus }(3,0) ; e=\frac{1}{2}$$

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

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}+4 x+y^{2}-8 y+32=0$$

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$\text { Focus }(0,-2) ; e=\frac{2}{3}$$

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

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$x^{2}-2 x+y^{2}+4 y=0$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$25 y^{2}-9 x^{2}=1$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$4 x^{2}+4 x+4 y^{2}-4 y-3=0$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{(x-1)^{2}}{9}-\frac{(y+3)^{2}}{25}=1$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$9 x^{2}+36 x+9 y^{2}=-32$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{(x+3)^{2}}{16}-\frac{(y-2)^{2}}{36}=1$$

Decide whether each equation has a circle as its graph. If it does, give the center and radius. $$9 x^{2}+9 y^{2}+54 y=-72$$

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$\text { Focus }(2,0) ; e=\frac{6}{5}$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{(y-5)^{2}}{4}-\frac{(x+1)^{2}}{9}=1$$

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 $$(x-4)^{2}=y+2$$

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$\frac{(y-5)^{2}}{4}-\frac{(x+1)^{2}}{9}=1$$

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 $$(x-2)^{2}=y+4$$