Chapter 8

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

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

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

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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$\frac{x^{2}}{16}-\frac{y^{2}}{9}=1$$

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

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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$\frac{y^{2}}{9}-\frac{x^{2}}{9}=1$$

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

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

Write an equation for each conic. Each parabola has vertex at the origin, and each ellipse or hyperbola is centered at the origin. $$\text { Focus }(0,8) ; e=1$$

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$49 y^{2}-36 x^{2}=1764$$

Firing a Projectile \(\quad\) A projectile is fired with an initial velocity of 400 feet per second at an angle of \(45^{\circ}\) with the horizontal. (See Example 5 ) (a) Find the time to the nearest tenth when it strikes the ground. (b) Find the range (horizontal distance covered). (c) What is the maximum altitude?

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

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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) 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. $$9 x^{2}+12 x+9 y^{2}-18 y-23=0$$

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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$\frac{4 x^{2}}{9}-\frac{25 y^{2}}{16}=1$$

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. $$x^{2}+2 x+y^{2}-6 y+14=0$$

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) 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. $$x^{2}+4 x+y^{2}-8 y+32=0$$

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$9 x^{2}-4 y^{2}=1$$

Alternative Parametric Forms 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. $$x^{2}-2 x+y^{2}+4 y=0$$

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$25 y^{2}-9 x^{2}=1$$

Alternative Parametric Forms Give two parametric representations of the parabola \(y=a(x-h)^{2}+k\)

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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) 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. $$9 x^{2}+36 x+9 y^{2}=-32$$

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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) 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. Vertical major axis of length \(6 ; e=\frac{4}{5}\)

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$\frac{(y-5)^{2}}{4}-\frac{(x+1)^{2}}{9}=1$$

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$(x-4)^{2}=y+2$$ 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

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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$\frac{(y+1)^{2}}{25}-\frac{(x-3)^{2}}{36}=1$$

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$(x-2)^{2}=y+4$$ 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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$16(x+5)^{2}-(y-3)^{2}=1$$

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$y+2=-(x-4)^{2}$$ 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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$4(x+9)^{2}-25(y+6)^{2}=100$$

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$y=-(x-2)^{2}-4$$ 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

CHECKING ANALYTIC SKILLS Graph each hyberbola by hand. Give the domain and range. Give the center in Exercises \(55-61 .\) Do not use a calculator. $$9(x-2)^{2}-4(y+1)^{2}=36$$

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$(y-4)^{2}=x+2$$ 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

CONCEPT CHECK 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.

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

Each equation in Exercises defines a parabola. Without actually graphing. match each equation with the appropriate description. $$x+2=-(y-4)^{2}$$ 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

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