Evaluate i 2 + 1 for all consecutive integers from 1 to 6, inclusive.Then find the sum

The set of all points in a plane that are equidistant from a fixed line and a fixed point is a/an . The fixed line is called the and the fixed point is called the .

Use the graph shown to answer Exercises 25. y x 2. The equation of the parabola is of the form a. y2 = 4px b. x2 = 4py.

If 4p = -28, then the coordinates of the focus are .

If 4p = -28, then the equation of the directrix is .

If 4p = -28, then the length of the latus rectum is . The endpoints of the latus rectum are and .

Use the graph shown to answer Exercises 69. y x (2, 1) 6. The equation of the parabola is of the form a. (y - 1) 2 = 4p(x + 2) b. (y + 1) 2 = 4p(x + 2) c. (x - 2) 2 = 4p(y - 1) d. (x + 2) 2 = 4p(y + 1).

If 4p = 4, then the coordinates of the focus are .

If 4p = 4, then the equation of the directrix is .

If 4p = 4, then the length of the latus rectum is . The endpoints of the latus rectum are and .

A nondegenerate conic section in the form Ax2 + Cy2 + Dx + Ey + F = 0, in which A and C are not both zero is a/an if A = C, a/an if AC = 0, a/an if A C and AC 7 0, and a/an if AC 6 0

In Exercises 516, find the focus and directrix of the parabola with the given equation. Then graph the parabola.x2 = -16y

In Exercises 516, find the focus and directrix of the parabola with the given equation. Then graph the parabola.x2 = -20y

In Exercises 516, find the focus and directrix of the parabola with the given equation. Then graph the parabola.y2 - 6x = 0

In Exercises 516, find the focus and directrix of the parabola with the given equation. Then graph the parabola.x2 - 6y = 0

In Exercises 516, find the focus and directrix of the parabola with the given equation. Then graph the parabola.8x2 + 4y = 0

In Exercises 516, find the focus and directrix of the parabola with the given equation. Then graph the parabola.8y2 + 4x = 0

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions. 17. Focus: (7, 0); Directrix: x = -7

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (9, 0); Directrix: x = -9

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (-5, 0); Directrix: x = 5

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (-10, 0); Directrix: x = 10

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (0, 15); Directrix: y = -15

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (0, 20); Directrix: y = -20

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (0, -25); Directrix: y = 25

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (0, -15); Directrix: y = 15

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Vertex: (2, -3); Focus: (2, -5)

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Vertex: (5, -2); Focus: (7, -2)

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (3, 2); Directrix: x = -1

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (2, 4); Directrix: x = -4

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (-3, 4); Directrix: y = 2

In Exercises 1730, find the standard form of the equation of each parabola satisfying the given conditions.Focus: (7, -1); Directrix: y = -9

In Exercises 3134, find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)(d). 31. (y - 1) 2 = 4(x - 1)

In Exercises 3134, find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)(d).(x + 1) 2 = 4(y + 1)

In Exercises 3134, find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)(d).(x + 1) 2 = -4(y + 1)

In Exercises 3134, find the vertex, focus, and directrix of each parabola with the given equation. Then match each equation to one of the graphs that are shown and labeled (a)(d).(y - 1) 2 = -4(x - 1)

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola. 35. (x - 2) 2 = 8(y - 1)

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola.

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola.

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola.

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola.

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola.(y + 4) 2 = 12(x + 2)

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola.(y - 1) 2 = -8x

In Exercises 3542, find the vertex, focus, and directrix of each parabola with the given equation. Then graph the parabola

In Exercises 4348, convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.x2 - 2x - 4y + 9 = 0

In Exercises 4348, convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.x2 + 6x + 8y + 1 = 0

In Exercises 4348, convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.y2 - 2y + 12x - 35 = 0

In Exercises 4348, convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.y2 - 2y - 8x + 1 = 0

In Exercises 4348, convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.x2 + 6x - 4y + 1 = 0

In Exercises 4348, convert each equation to standard form by completing the square on x or y. Then find the vertex, focus, and directrix of the parabola. Finally, graph the parabola.x2 + 8x - 4y + 8 = 0

In Exercises 4956, identify each equation without completing the square. 49. y2 - 4x + 2y + 21 = 0

In Exercises 4956, identify each equation without completing the square.y2 - 4x - 4y = 0

In Exercises 4956, identify each equation without completing the square.4x2 - 9y2 - 8x - 36y - 68 = 0

In Exercises 4956, identify each equation without completing the square.9x2 + 25y2 - 54x - 200y + 256 = 0

In Exercises 4956, identify each equation without completing the square.4x2 + 4y2 + 12x + 4y + 1 = 0

In Exercises 4956, identify each equation without completing the square.9x2 + 4y2 - 36x + 8y + 31 = 0

In Exercises 4956, identify each equation without completing the square.100x2 - 7y2 + 90y - 368 = 0

In Exercises 4956, identify each equation without completing the square.y2 + 8x + 6y + 25 = 0

In Exercises 5762, use the vertex and the direction in which the parabola opens to determine the relations domain and range. Is the relation a function? 57. y2 + 6y - x + 5 = 0

In Exercises 5762, use the vertex and the direction in which the parabola opens to determine the relations domain and range. Is the relation a function?

In Exercises 5762, use the vertex and the direction in which the parabola opens to determine the relations domain and range. Is the relation a function?

In Exercises 5762, use the vertex and the direction in which the parabola opens to determine the relations domain and range. Is the relation a function?

In Exercises 5762, use the vertex and the direction in which the parabola opens to determine the relations domain and range. Is the relation a function?

In Exercises 5762, use the vertex and the direction in which the parabola opens to determine the relations domain and range. Is the relation a function?

In Exercises 6368, find the solution set for each system by graphing both of the systems equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.c (y - 2) 2 = x + 4 y = - 1 2 x

In Exercises 6368, find the solution set for each system by graphing both of the systems equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.b (y - 3) 2 = x - 2 x + y = 5

In Exercises 6368, find the solution set for each system by graphing both of the systems equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.b x = y2 - 3 x = y2 - 3y

In Exercises 6368, find the solution set for each system by graphing both of the systems equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.b x = y2 - 5 x2 + y2 = 25

In Exercises 6368, find the solution set for each system by graphing both of the systems equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.b x = (y + 2) 2 - 1 (x - 2) 2 + (y + 2) 2 = 1

In Exercises 6368, find the solution set for each system by graphing both of the systems equations in the same rectangular coordinate system and finding points of intersection. Check all solutions in both equations.b x = 2y2 + 4y + 5 (x + 1) 2 + (y - 2) 2 = 1

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?

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?

A satellite dish, like the one shown below, is in the shape of a parabolic surface. Signals coming from a satellite strike the surface of the dish and are reflected to the focus, where the receiver is located. The satellite dish shown has a diameter of 12 feet and a depth of 2 feet. How far from the base of the dish should the receiver be placed?

In Exercise 71, if the diameter of the dish is halved and the depth stays the same, how far from the base of the smaller dish should the receiver be placed?

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.

The towers of a suspension bridge are 800 feet apart and rise 160 feet 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 100 feet from a tower?

The parabolic arch shown in the figure is 50 feet above the water at the center and 200 feet wide at the base. Will a boat that is 30 feet tall clear the arch 30 feet from the center?

A satellite dish in the shape of a parabolic surface has a diameter of 20 feet. If the receiver is to be placed 6 feet from the base, how deep should the dish be?

What is a parabola?

Explain how to use y2 = 8x to find the parabolas focus and directrix

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 one similarity and one difference between the graphs of y2 = 4x and (y - 1) 2 = 4(x - 1).

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

Look at the satellite dish shown in Exercise 71.Why must the receiver for a shallow dish be farther from the base of the dish than for a deeper dish of the same diameter?

Explain how to identify the graph of Ax2 + Cy2 + Dx + Ey + F = 0.

Use a graphing utility to graph any five of the parabolas that you graphed by hand in Exercises 516.

Use a graphing utility to graph any three of the parabolas that you graphed by hand in Exercises 3542. First solve the given equation for y, possibly using the square root property.

Use a graphing utility to graph the parabolas in Exercises 8687. 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. 86. y2 + 2y - 6x + 13 = 0

Use a graphing utility to graph the parabolas in Exercises 8687. 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.y2 + 10y - x + 25 = 0

In Exercises 8889, write each equation as a quadratic equation in y and then use the quadratic formula to express y in terms of x. Graph the resulting two equations using a graphing utility. What effect does the xy-term have on the graph of the resulting parabola?16x2 - 24xy + 9y2 - 60x - 80y + 100 = 0

In Exercises 8889, write each equation as a quadratic equation in y and then use the quadratic formula to express y in terms of x. Graph the resulting two equations using a graphing utility. What effect does the xy-term have on the graph of the resulting parabola?x2 + 223xy + 3y2 + 823x - 8y + 32 = 0

In Exercises 9093, determine whether each statement makes sense or does not make sense, and explain your reasoning. 90. I graphed a parabola that opened to the right and contained a maximum point

Knowing that a parabola opening to the right has a vertex at (-1, 1) gives me enough information to determine its graph

I noticed that depending on the values for A and C, assuming that they are not both zero, the graph of Ax2 + Cy2 + Dx + Ey + F = 0 can represent any of the conic sections.

Im using a telescope in which light from distant stars is reflected to the focus of a parabolic mirror.

In Exercises 9497, determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. 94. The parabola whose equation is x = 2y - y2 + 5 opens to the right.

If the parabola whose equation is x = ay2 + by + c has its vertex at (3, 2) and a 7 0, then it has no y@intercepts.

Some parabolas that open to the right have equations that define y as a function of x.

The graph of x = a(y - k) + h is a parabola with vertex at (h, k).

Find the focus and directrix of a parabola whose equation is of the form Ax2 + Ey = 0, A 0, E 0.

Write the standard form of the equation of a parabola whose points are equidistant from y = 4 and (-1, 0).

Consult the research department of your library or the Internet to find an example of architecture that incorporates one or more conic sections in its design. Share this example with other group members. Explain precisely how conic sections are used. Do conic sections enhance the appeal of the architecture? In what ways?

Solve the system: e y = x2 - 7 x2 + y2 = 13. (Section 5.4, Example 4)

. Consider the system c x - y + z = -3 -2y + z = -6 -2x - 3y = -10. a. Write the system as a matrix equation in the form AX = B. b. Solve the system using the fact that the inverse of C 1 -1 1 0 -2 1 -2 -3 0 S is C 3 -3 1 -2 2 -1 -4 5 -2 S . (Section 6.4, Example 5)

Use Cramers Rule (determinants) to solve the system: e x - y = -5 3x + 2y = 0. (Section 6.5, Example 2)

Exercises 104106 will help you prepare for the material covered in the first section of the next chapter. 104. Evaluate (-1) n 3n - 1 for n = 1, 2, 3, and 4

Find the product of all positive integers from n down through 1 for n = 5.

Evaluate i 2 + 1 for all consecutive integers from 1 to 6, inclusive.Then find the sum of the six evaluations.