Chapter 7

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y^{2}-2 y-x-5=0$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. \(9 x^{2}-16 y^{2}-36 x-64 y+116=0\)

Graph each ellipse and give the location of its foci. $$36(x+4)^{2}+(y+3)^{2}=36$$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y=-x^{2}+4 x-3$$

Graph each relation. Use the relation's graph to determine its domain and range. \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\)

Convert each equation to standard form by ecompleting the square on \(x\) and \(y .\) Then graph the ellipse and give ehe location of its foci. $$9 x^{2}+25 y^{2}-36 x+50 y-164=0$$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$y=-x^{2}-4 x+4$$

Convert each equation to standard form by ecompleting the square on \(x\) and \(y .\) Then graph the ellipse and give ehe location of its foci. $$4 x^{2}+9 y^{2}-32 x+36 y+64=0$$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=-4(y-1)^{2}+3$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{r}\frac{x^{2}}{25}+\frac{y^{2}}{9}=1 \\\y=3\end{array}\right.$$

Graph each relation. Use the relation's graph to determine its domain and range. \(\frac{x^{2}}{9}+\frac{y^{2}}{16}=1\)

Convert each equation to standard form by ecompleting the square on \(x\) and \(y .\) Then graph the ellipse and give ehe location of its foci. $$9 x^{2}+16 y^{2}-18 x+64 y-71=0$$

Use the vertex and the direction in which the parabola opens to determine the relation's domain and range. Is the relation a function? $$x=-3(y-1)^{2}-2$$

Convert each equation to standard form by ecompleting the square on \(x\) and \(y .\) Then graph the ellipse and give ehe location of its foci. $$x^{2}+4 y^{2}+10 x-8 y+13=0$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{array}{r} (y-2)^{2}=x+4 \\ y=-\frac{1}{2} x \end{array}\right. $$

Graph each relation. Use the relation's graph to determine its domain and range. \(\frac{y^{2}}{16}-\frac{x^{2}}{9}=1\)

Convert each equation to standard form by ecompleting the square on \(x\) and \(y .\) Then graph the ellipse and give ehe location of its foci. $$4 x^{2}+y^{2}+16 x-6 y-39=0$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{array}{c} (y-3)^{2}=x-2 \\ x+y=5 \end{array}\right. $$

Convert each equation to standard form by ecompleting the square on \(x\) and \(y .\) Then graph the ellipse and give ehe location of its foci. $$4 x^{2}+25 y^{2}-24 x+100 y+36=0$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{l}x^{2}+y^{2}-1 \\\x^{2}+9 y^{2}-9\end{array}\right.$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.$$\left\\{\begin{array}{c}x^{2}+y^{2}-25 \\\25 x^{2}+y^{2}-25\end{array}\right.$$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{array}{l} x=(y+2)^{2}-1 \\ (x-2)^{2}+(y+2)^{2}=1 \end{array}\right. $$

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$ \left\\{\begin{aligned} \frac{x^{2}}{4}+\frac{y^{2}}{36} &-1 \\ x &\--2 \end{aligned}\right. $$

The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 4 inches and a depth of 1 inch. How far from the vertex should the light bulb be placed?

An explosion is recorded by two microphones that are 1 mile eqpart. Microphone \(M_{1}\) received the sound 2 seconds before microphone \(M_{2} .\) Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones.

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{c}4 x^{2}+y^{2}=4 \\\2 x-y=2\end{array}\right.$$

The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 8 inches and a depth of 1 inch. How far from the vertex should the light bulb be placed?

Radio towers \(A\) and \(B, 200\) kilometers apart, are situated along the coast, with \(A\) located due west of \(B\). Simultaneous radio signals are sent from each tower to a ship, with the signal from \(B\) received 500 microseconds before the signal from \(A\) a. Assuming that the radio signals travel 300 meters per microsecond, determine the equation of the hyperbola on which the ship is located. b. If the ship lies due north of tower \(B\), how far out at sea is it?

Find the solution set for each system by graphing both of the system's equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations. $$\left\\{\begin{array}{c}4 x^{2}+y^{2}=4 \\\x+y=3\end{array}\right.$$

Graph each semi ellipse. $$y=-\sqrt{16-4 x^{2}}$$

Graph each semi ellipse. $$y=-\sqrt{4-4 x^{2}}$$

The towers of the Golden Gate Bridge connecting San Francisco to Marin County are 1280 meters apart and rise 160 meters above the road. The cable between the towers has the shape of a parabola and the cable just touches the sides of the road midway between the towers. What is the height of the cable 200 meters from a tower? Round to the nearest meter. (Image can't copy)

What is a hyperbola?

Describe how to graph \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\)

What is a parabola?

What is an ellipse?

Describe how to locate the foci of the graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{y^{2}}{9}-\frac{x^{2}}{1}=1\)

Explain how to use \(y^{2}=8 x\) to find the parabola's focus and directrix.

Describe how to graph \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\)

If you are given the standard form of the equation of a parabola with vertex at the origin, explain how to determine if the parabola opens to the right, left, upward, or downward.

Describe how to locate the foci for \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\)

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{9}-\frac{y^{2}}{1}=1\) and \(\frac{(x-3)^{2}}{9}-\frac{(y+3)^{2}}{1}=1\)

Describe one similarity and one difference between the graphs of \(y^{2}=4 x\) and \((y-1)^{2}=4(x-1)\)

How can you distinguish an ellipse from a hyperbola by looking at their equations?

How can you distinguish parabolas from other conic sections by looking at their equations?

Describe one similarity and one difference between the graphs of \(\frac{x^{2}}{25}+\frac{y^{2}}{16}=1\) and \(\frac{(x-1)^{2}}{25}+\frac{(y-1)^{2}}{16}=1\)

In \(1992,\) a NASA team began a project called Spaceguard Survey, calling for an international watch for comets that might collide with Earth. Why is it more difficult to detect a possible "doomsday comet" with a hyperbolic orbit than one with an elliptical orbit?

An elliptipool is an elliptical pool table with only one pocket. A pool shark places a ball on the table, hits it in what appears Fo be a random direction, and yet it bounces off the edge, Elalling directly into the pocket. Explain why this happens.

Use a graphing utility to graph the parabolas in Exercises 77-78. Write the given equation as a quadratic equation in \(y\) and use the quadratic formula to solve for \(y .\) Enter each of the equations to produce the complete graph. $$y^{2}+2 y-6 x+13=0$$