Chapter 7

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y=-3 x^{2}+24 x-46$$

Use the definition of an ellipse to find an equation of an ellipse with foci \((3,0)\) and \((-3,0),\) where the sum of the distances from any point of the ellipse to the two foci is 10

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=y^{2}+2$$

Use the definition of a hyperbola to find an equation of hyperbola with center at the origin, foci \((-2,0)\) and \(=2,0),\) and the absolute value of the difference of the Jistances from any point of the hyperbola to the two foci equal to 2

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=(y+1)^{2}$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=(y-3)^{2}$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$(y+2)^{2}=x+1$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=(y-4)^{2}+2$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-2(y+3)^{2}$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=\frac{2}{3} y^{2}-4 y+8$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=y^{2}+2 y-8$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-4 y^{2}-4 y-3$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-2 y^{2}+2 y-3$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=2 y^{2}-4 y+6$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$2 x=y^{2}-4 y+6$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$2 x=y^{2}-2 y+9$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$x=-3 y^{2}+6 y-1$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y^{2}-4 y+4=4 x+4$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. $$y^{2}+2 y+1=-2 x+4$$

Graph each parabola by hand, and check using a graphing calculator. Give the vertex, axis, domain, and range. Prove that the parabola with focus \((c, 0)\) and directrix \(x=-c\) has equation \(y^{2}=4 c x\)

Path of an Object on a Planet When an object moves under the influence of a gravitational force (without air resistance), its path can be parabolic. This is the path of a ball thrown near the surface of a planet or other celestial object. Suppose two balls are simultaneously thrown upward at a \(45^{\circ}\) angle on two different planets. If their initial velocities are both \(30 \mathrm{mph}\), then their \(x y\) -coordinates in feet can be expressed by the equation $$ y=x-\frac{g}{1922} x^{2} $$ where \(g\) is the acceleration due to gravity. The value of \(g\) will vary with the mass and size of the planet. (Source: Zeilik, M., S. Gregory, and E. Smith, Introductory Astronomy and Astrophysics, Saunders College Publishers.) (a) On Earth, \(g=32.2\) and on Mars, \(g=12.6 .\) Find the two equations, and use the same screen of a graphing calculator to graph the paths of the two balls thrown on Earth and Mars. Use the window [0,180] by \([0,120] .\) (Hint: If possible, set the mode on your graphing calculator to simultaneous.) (b) Determine the difference in the horizontal distances traveled by the two balls.

Design of a Radio Telescope The U.S. Naval Research Laboratory designed a giant radio telescope weighing 3450 tons. Its parabolic dish had a diameter of 300 feet, with a focal length (the distance from the focus to the parabolic surface) of 128.5 feet. Determine the maximum depth of the 300 -foot dish. (Source: Mar, J. and H. Liebowitz, Structure Technology for Large Radio and Radar Telescope Systems, MIT Press.)

Particle When an alpha particle (a subatomic particle) is moving in a horizontal path along the positive \(x\) -axis and passes between charged plates, it is deflected in a parabolic path. If the plate is charged with 2000 volts and is 0.4 meter long, then an alpha particle's path can be described by the equation \(y=-\frac{k}{2 v_{0}} x^{2}\) where \(k=5 \times 10^{-9}\) is constant and \(v_{0}\) is the initial velocity of the particle. If \(v_{0}=10^{7}\) meters per second, what is the deflection of the alpha particle's path in the \(y\) -direction when \(x=0.4\) meter? (Source: Semat, H. and J. Albright, Introduction to Atomic and Nuclear Physics, Holt, Rinehart and Winston.)

Headlight \(A\) headlight is being constructed in the shape of a paraboloid with depth 4 inches and diameter 5 inches, as illustrated in the figure. Determine the distance \(d\) that the bulb should be from the vertex in order to have the beam of light shine straight ahead. (IMAGE CANNOT COPY)