Solved: Path of a Softball The path of a softball is modeled by where the coordinates

A ________ is the intersection of a plane and a double-napped cone.

When a plane passes through the vertex of a double-napped cone, the intersection is a ________ ________.

A collection of points satisfying a geometric property can also be referred to as a ________ of points.

A ________ is defined as the set of all points in a plane that are equidistant from a fixed line, called the ________, and a fixed point, called the ________, not on the line.

The line that passes through the focus and the vertex of a parabola is called the ________ of the parabola.

The ________ of a parabola is the midpoint between the focus and the directrix.

A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a ________ ________.

A line is ________ to a parabola at a point on the parabola when the line intersects, but does not cross, the parabola at the point.

y 2  4x

x 2  2y

x 2  8y

y 2  12x

y 12  4x 3 y 2

x 32  2y 1 y 12

4 2 2 4 2 4 6 (3, 6) y

8 4 4 8 8 (2, 6) y

Focus: 0, , 0 1 2 x

Focus: 3 2 0, , 0 1 2

Focus: 2, 0 0, 2

Focus: 0, 2

Directrix: y  1 y

Directrix: y  2

Directrix: x  1 x

Directrix: x  3

Vertical axis and passes through the point 4, 6

Vertical axis and passes through the point 3, 3

Horizontal axis and passes through the point 2, 5

Horizontal axis and passes through the point 3, 2

y  1 2x 2

y  2x 2 y

y 2  3x 2  6x

y y 2  3x

x 2  0 2 6y  0

x y x 2  0 2

x 1 2 8y 2  0 x y x

x 5 y 1 2  0 x 1 2

x 3  4y 1 2  4 y 3 2 x 5 y

x 1 2 2 x 3  4y 1 2  4 y

y  2 2y 33 1 4x 2 2x 5 x 1

x  1 4y y  2 2y 33 1 4x

y 2 4y 4x  0 2 6y 8x 25  0 x

y y 2 4y 4x  0 2

x 2 2x 8y 9  0 2 4x 6y 2  0 Copyr

x x 2 2x 8y 9  0 2

y 2 4x 4  0 2 x y  0 x x

y y 2 4x 4  0 2

y (2, 0)

(4.5, 4) (5, 3) y

( 4, 0) y

8 12 4 (3, 3) (0, 0) y

Vertex: focus:

Vertex: focus:

Vertex: directrix:

Vertex: directrix:

Focus: directrix:

Focus: directrix:

x 4, 8 2

x x 4, 8 2  2y, 2

2x 1, 2 , 2,

2x 2, 8 2

Highway Design Highway engineers design a parabolic curve for an entrance ramp from a straight street to an interstate highway (see figure). Find an equation of the parabola.

Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). (a) Find an equation of the parabola with its vertex at the origin that models the road surface. (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

Flashlight The light bulb in a flashlight is at the focus of a parabolic reflector, 1.5 centimeters from the vertex of the reflector (see figure). Write an equation for a cross section of the flashlights reflector with its focus on the positive -axis and its vertex at the origin.

Satellite Antenna Write an equation for a cross section of the parabolic satellite dish antenna shown in the figure.

Beam Deflection A simply supported beam is 64 feet long and has a load at the center (see figure). The deflection (bending) of the beam at its center is 1 inch. The shape of the deflected beam is parabolic. (a) Find an equation of the parabola with its vertex at the origin that models the shape of the beam. (b) How far from the center of the beam is the deflection equal to inch?

Beam Deflection Repeat Exercise 65 when the length of the beam is 36 feet and the deflection of the beam at its center is 2 inches.

Fluid Flow Water is flowing from a horizontal pipe 48 feet above the ground. The falling stream of water has the shape of a parabola whose vertex is at the end of the pipe (see figure). The stream of water strikes the ground at the point Find the equation of the path taken by the water.

Window Design A church window (see figure) is bounded above by a parabola. Find the equation of the parabola

Archway A parabolic archway (see figure) is 12 meters high at the vertex. At a height of 10 meters, the width of the archway is 8 meters. How wide is the archway at ground level?

Lattice Arch A parabolic lattice arch is 8 feet high at the vertex. At a height of 4 feet, the width of the lattice arch is 4 feet (see figure). How wide is the lattice arch at ground level?

Suspension Bridge Each cable of a suspension bridge is suspended (in the shape of a parabola) between two towers (see figure). (a) Find the coordinates of the focus. (b) Write an equation that models the cables.

Suspension Bridge Each cable of the Golden Gate Bridge is suspended (in the shape of a parabola) between two towers that are 1280 meters apart. The top of each tower is 152 meters above the roadway. The cables touch the roadway midway between the towers. (a) Draw a sketch of the bridge. Locate the origin of a rectangular coordinate system at the center of the roadway. Label the coordinates of the known points. (b) Write an equation that models the cables. (c) Complete the table by finding the height of the suspension cables over the roadway at a distance of meters from the center of the bridge.

Weather Satellite Orbit A weather satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. When this velocity is multiplied by the weather satellite has the minimum velocity necessary to escape Earths gravity and follow a parabolic path with the center of Earth as the focus (see figure). (a) Find the escape velocity of the weather satellite. (b) Find an equation of the parabolic path of the weather satellite. (Assume that the radius of Earth is 4000 miles.)

Path of a Softball The path of a softball is modeled by where the coordinates and are measured in feet, with corresponding to the position from which the ball was thrown. (a) Use a graphing utility to graph the trajectory of the softball. (b) Use the trace feature of the graphing utility to approximate the highest point and the range of the trajectory

A ball is thrown from the top of a 100-foot tower with a velocity of 28 feet per second. (a) Find the equation of the parabolic path. (b) How far does the ball travel horizontally before striking the ground?

A cargo plane is flying at an altitude of 500 feet and a speed of 135 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel horizontally before it hits the ground?

It is possible for a parabola to intersect its directrix.

If the vertex and focus of a parabola are on a horizontal line, then the directrix of the parabola is vertical.

Slope of a Tangent Line Let be the coordinates of a point on the parabola The equation of the line tangent to the parabola at the point is What is the slope of the tangent line?

Think About It Explain what each of the following equations represents, and how equations (a) and (b) are equivalent.

Think About It The equation is a degenerate conic. Sketch the graph of this equation and identify the degenerate conic. Describe the intersection of the plane and the double-napped cone for this particular conic.

HOW DO YOU SEE IT? In parts (a)(d), describe how a plane could intersect the double-napped cone to form the conic section (see figure). (a) Circle (b) Ellipse (c) Parabola (d) Hyperbola

Graphical Reasoning Consider the parabola (a) Use a graphing utility to graph the parabola for and Describe the effect on the graph when increases. (b) Locate the focus for each parabola in part (a). (c) For each parabola in part (a), find the length of the latus rectum (see figure). How can the length of the latus rectum be determined directly from the standard form of the equation of the parabola? (d) Explain how the result of part (c) can be used as a sketching aid when graphing parabolas.

Geometry The area of the shaded region in the figure is (a) Find the area when and (b) Give a geometric explanation of why the area approaches 0 as approaches 0. p p  2 b  4. y = b x y x 2 = 4py A  8 3 p12 b32.