Chapter 8

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. $$\begin{aligned} &x=t^{2}, y=\sqrt{t}, \text { for } t \text { in }[0,4]\\\ &\text { window: }\lfloor- 2,20\rfloor \text { by }\lfloor 0,4\rfloor \end{aligned}$$

Graph each ellipse by hand. Give the domain and range. Give the foci in Exercises \(11-14\) and identify the center in Exercises \(17-22 .\) Do not use a calculator. $$\frac{16 y^{2}}{9}+\frac{121 x^{2}}{25}=1$$

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}=25-y^{2}$$

Find the center-radius form for each circle satisfying the given conditions. Center \((2,-7) ;\) passing through \((-2,-4)\)

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. $$\begin{aligned} &x=t^{3}+1, y=t^{3}-1, \text { for } t \text { in }[-3,3]\\\ &\text { window: }[-30,30] \text { by }[-30,30] \end{aligned}$$

Graph each ellipse by hand. Give the domain and range. Give the foci in Exercises \(11-14\) and identify the center in Exercises \(17-22 .\) Do not use a calculator. $$\frac{(x-1)^{2}}{9}+\frac{(y+3)^{2}}{25}=1$$

Find the center-radius form for each circle satisfying the given conditions. Center \((-3,-2) ;\) tangent to the \(x\) -axis (Hint: "tangent to" means touching at one point.)

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=2 t-1, y=t^{2}+2,\) for \(t\) in \([-10,10]\)window: \([-20,20]\) by \([0,120]\)

Graph each ellipse by hand. Give the domain and range. Give the foci in Exercises \(11-14\) and identify the center in Exercises \(17-22 .\) Do not use a calculator. $$\frac{(x+3)^{2}}{16}+\frac{(y-2)^{2}}{36}=1$$

Find the center-radius form for each circle satisfying the given conditions. Center \((5,-1) ;\) tangent to the \(y\) -axis

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=2^{t}, y=\sqrt{3 t-1},\) for \(t\) in \(\left[\frac{1}{3}, 4\right]\)window: \([-2,30]\) by \([-2,10]\)

Graph each ellipse by hand. Give the domain and range. Give the foci in Exercises \(11-14\) and identify the center in Exercises \(17-22 .\) Do not use a calculator. $$\frac{(x-2)^{2}}{16}+\frac{(y-1)^{2}}{9}=1$$

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$4(x-3)^{2}+3(y+4)^{2}=0$$

Describe the graph of the following equation. $$(x-3)^{2}+(y-3)^{2}=0$$

Graph each ellipse by hand. Give the domain and range. Give the foci in Exercises \(11-14\) and identify the center in Exercises \(17-22 .\) Do not use a calculator. $$\frac{(x+3)^{2}}{25}+\frac{(y+2)^{2}}{36}=1$$

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x-4 y^{2}-8 y=0$$

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=t+2, y=-\frac{1}{2} \sqrt{9-t^{2}},\) for \(t\) in \([-3,3]\)window: \([-6,6]\) by \([-4,4]\)

Graph each ellipse by hand. Give the domain and range. Give the foci in Exercises \(11-14\) and identify the center in Exercises \(17-22 .\) Do not use a calculator. $$\frac{(x+1)^{2}}{64}+\frac{(y-2)^{2}}{49}=1$$

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$2 x^{2}-8 x+2 y^{2}+20 y=12$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius which is the distance from the center to either endpoint of the diameter. Finally, use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-1,3) \text { and }(5,-9)$$

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=t, y=\sqrt{4-t^{2}},\) for \(t\) in \([-2,2]\)window: \([-6,6]\) by \([-4,4]\)

Graph each ellipse by hand. Give the domain and range. Give the foci in Exercises \(11-14\) and identify the center in Exercises \(17-22 .\) Do not use a calculator. $$\frac{(x-4)^{2}}{9}+\frac{(y+2)^{2}}{4}=1$$

The equation of a conic section is given in a familiar form. Identify the type of graph (if any) that each equation has, without actually graphing. See the summary chart in this section. Do not use a calculator. $$x^{2}+2 x=-4 y$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius which is the distance from the center to either endpoint of the diameter. Finally, use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-4,5) \text { and }(6,-9)$$

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=t, y=\frac{1}{t},\) for \(t\) in \([-10,10]\)window: \([-6,6]\) by \([-4,4]\)

Find an equation for each ellipse. \(x\) -intercepts \((\pm 4,0) ;\) foci \((\pm 2,0)\)

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius which is the distance from the center to either endpoint of the diameter. Finally, use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-5,-7) \text { and }(1,1)$$

Use a graphing calculator to generate the curve over the interval for the parameter \(t\), in the window specified. Then find a rectangular equation for the curve. \(x=2 t-1, y=\frac{1}{t},\) for \(t\) in \([-10,10]\)window: \([-6,6]\) by \([-4,4]\)

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

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius which is the distance from the center to either endpoint of the diameter. Finally, use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-3,-2) \text { and }(1,-4)$$

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t, y=t-1, \text { for } t \text { in }(-\infty, \infty)$$

Find an equation for each ellipse. \(y\) -intercepts \((0, \pm 2 \sqrt{2}) ;\) foci \((0, \pm 2)\)

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$x^{2}=4 y-8$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius which is the distance from the center to either endpoint of the diameter. Finally, use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(-5,0) \text { and }(5,0)$$

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t+3, y=2 t, \text { for } t \text { in }(-\infty, \infty)$$

Find an equation for each ellipse. \(x\) -intercepts \((\pm 3 \sqrt{2}, 0) ;\) foci \((\pm 2 \sqrt{3}, 0)\)

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$\frac{x^{2}}{4}+\frac{y^{2}}{4}=1$$

We can find an equation of a circle if we know the coordinates of the endpoints of a diameter of the circle. First, find the midpoint of the diameter, which is the center of the circle. Then find the radius which is the distance from the center to either endpoint of the diameter. Finally, use the center-radius form to find the equation. Find the center-radius form for each circle having the given endpoints of a diameter. $$(0,9) \text { and }(0,-9)$$

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t^{2}, y=t+1, \text { for } t \text { in }(-\infty, \infty)$$

Find an equation for each ellipse. \(x\) -intercepts \((\pm 4,0) ; y\) -intercepts \((0, \pm 2)\)

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$x^{2}=25+y^{2}$$

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=t-2, y=\frac{1}{2} t^{2}+1, \text { for } t \text { in }(-\infty, \infty)$$

Find an equation for each ellipse. \(x\) -intercepts \((\pm 3,0) ; y\) -intercepts \((0, \pm 6)\)

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$9 x^{2}+36 y^{2}=36$$

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=3 t^{2}, y=4 t^{3}, \text { for } t \text { in }(-\infty, \infty)$$

Find an equation for each ellipse. Endpoints of major axis at \((6,0)\) and \((-6,0) ; c=4\)

Graph each circle by hand if possible. Give the domain and range. $$x^{2}+y^{2}=4$$

Find a rectangular equation. State the appropriate interval for \(x\) or \(y .\) $$x=2 t^{3}, y=-t^{2}, \text { for } t \text { in }(-\infty, \infty)$$

Find an equation for each ellipse. Vertices \((0,5)\) and \((0,-5) ; b=2\)

Determine the type of conic section represented by each equation, and graph it, provided a graph exists. $$\frac{(x-4)^{2}}{8}+\frac{(y+1)^{2}}{2}=0$$