In Exercises 53 and 54, the equations of a parabola and a tangent line to the parabola

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

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

Fill in the blanks. 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.

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

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

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

Fill in the blanks. A line is ________ to a parabola at a point on the parabola if the line intersects, but does not cross, the parabola at the point.

In Exercises 510, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]

In Exercises 510, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]

In Exercises 510, match the equation with its graph. [The graphs are labeled (a), (b), (c), (d), (e), and (f).]

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 1124, find the vertex, focus, and directrix of the parabola and sketch its graph.

In Exercises 2528, find the vertex, focus, and directrix of the parabola.Use a graphing utility to graph the parabola.

In Exercises 2528, find the vertex, focus, and directrix of the parabola.Use a graphing utility to graph the parabola.

In Exercises 2528, find the vertex, focus, and directrix of the parabola.Use a graphing utility to graph the parabola.

In Exercises 2528, find the vertex, focus, and directrix of the parabola.Use a graphing utility to graph the parabola. y2 _ 4x _ 4 _ 0

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin.

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin.

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Focus:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Directrix:

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Horizontal axis and passes through the point

In Exercises 2940, find the standard form of the equation of the parabola with the given characteristic(s) and vertex at the origin. Vertical axis and passes through the point

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics.

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics.

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics.

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics.

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics. Vertex: focus:

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics. Vertex: focus:

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics. Vertex: directrix:

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics. Vertex: directrix:

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics. Focus: directrix:

In Exercises 4150, find the standard form of the equation of the parabola with the given characteristics. Focus: directrix:

In Exercises 51 and 52,change the equation of the parabola so that its graph matches the description. upper half of parabola

In Exercises 51 and 52,change the equation of the parabola so that its graph matches the description. lower half of parabola Parabola Tangent Line

In Exercises 53 and 54, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency.

In Exercises 53 and 54, the equations of a parabola and a tangent line to the parabola are given. Use a graphing utility to graph both equations in the same viewing window. Determine the coordinates of the point of tangency.

In Exercises 5558, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line

In Exercises 5558, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line

In Exercises 5558, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line

In Exercises 5558, find an equation of the tangent line to the parabola at the given point, and find the -intercept of the line

Revenue The revenue (in dollars) generated by the sale of units of a patio furniture set is given by Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue.

Revenue The revenue (in dollars) generated by the sale of units of a digital camera is given by Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue.

Satellite Antenna The receiver in a parabolic television dish antenna is 4.5 feet from the vertex and is located at the focus (see figure). Write an equation for a cross section of the reflector. (Assume that the dish is directed upward and the vertex is at the origin.) 4.5 ft Receiver x y

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 x meters from the center of the bridge.

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). Cross section of road surface (a) Find an equation of the parabola that models the road surface. (Assume that the origin is at the center of the road.) (b) How far from the center of the road is the road surface 0.1 foot lower than in the middle?

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. FIGURE FOR 64

Satellite Orbit A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by the satellite will have the minimum velocity necessary to escape Earths gravity and it will follow a parabolic path with the center of Earth as the focus (see figure). (a) Find the escape velocity of the satellite. (b) Find an equation of the parabolic path of the 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. Projectile Motion X

A ball is thrown from the top of a 75-foot tower with a velocity of 32 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 30,000 feet and a speed of 540 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel horizontally before it hits the ground? True or False?

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.

Exploration 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 chord passing through the focus and parallel to the directrix (see figure). How can the length of this chord 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.

Exploration 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?

Writing In your own words, state the reflective property of a parabola.

In Exercises 7578, list the possible rational zeros of given by the Rational Zero Test.

In Exercises 7578, list the possible rational zeros of given by the Rational Zero Test.

In Exercises 7578, list the possible rational zeros of given by the Rational Zero Test.

In Exercises 7578, list the possible rational zeros of given by the Rational Zero Test.

Find a polynomial with real coefficients that has the zeros and

Find all the zeros of if one of the zeros is

Find all the zeros of the function if two of the zeros are

Use a graphing utility to graph the function given by Use the graph to approximate the zeros of

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places.

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places.

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places.

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places.

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places.

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places.

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places.

In Exercises 8390, use the information to solve the triangle. Round your answers to two decimal places. B _ 71_, a _ 21, c _ 29