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7th Edition

James Stewart + 2 others

Publisher: Brooks Cole

ISBN: 9781305071759

Chapter 11, Problem 9RE

Textbook Problem

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**Graphing Parabolas** An equation of a parabola is given. **(a)** Find the vertex, focus, and directrix of the parabola. **(b)** Sketch a graph of the parabola and its directrix.

**9.**

(a)

To determine

**To find:** The vertex, focus and directrix of the parabola

**Definition used:**

“The equation of shifted parabola with vertex

“The equation of the parabola with vertex

“If

The given equation of conic is

Given the equation of conic

Therefore, the given conic is

Use the definition “The equation of shifted parabola with vertex

Here,

Therefore, by above definition the vertex of the parabola

The equation

(b)

To determine

**To sketch:** The graph and directrix of the parabola

Precalculus: Mathematics for Calculus (Standalone Book)

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Ch. 11.1 - A parabola is the set of all points in the plane...Ch. 11.1 - The graph of the equation x2 = 4py is a parabola...Ch. 11.1 - The graph of the equation y2 = 4px is a parabola...Ch. 11.1 - Label the focus, directrix, and vertex on the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...Ch. 11.1 - Graphs of Parabolas Match the equation with the...

Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas An equation of a parabola is...Ch. 11.1 - Graphing Parabolas Use a graphing device to graph...Ch. 11.1 - Graphing Parabolas Use a graphing device to graph...Ch. 11.1 - Graphing Parabolas Use a graphing device to graph...Ch. 11.1 - 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A latus rectum...Ch. 11.2 - Perihelion and Aphelion The planets move around...Ch. 11.2 - The Orbit of Pluto With an eccentricity of 0.25,...Ch. 11.2 - Lunar Orbit For an object in an elliptical orbit...Ch. 11.2 - Plywood Ellipse A carpenter wishes to construct an...Ch. 11.2 - Sunburst Window A sunburst window above a doorway...Ch. 11.2 - DISCUSS: Drawing an Ellipse on a Blackboard Try...Ch. 11.2 - DISCUSS: Light Cone from a Flashlight A flashlight...Ch. 11.2 - DISCUSS: Is It an Ellipse? 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Conjugate Hyperbolas The hyperbolas...Ch. 11.3 - Equation of a Hyperbola In the derivation of the...Ch. 11.3 - Verifying a Geometric Property of a Hyperbola (a)...Ch. 11.3 - Confocal Hyperbolas Hyperbolas are called confocal...Ch. 11.3 - Navigation In the figure on the next page, the...Ch. 11.3 - Comet Trajectories Some comets, such as Halleys...Ch. 11.3 - Ripples in Pool Two stones are dropped...Ch. 11.3 - DISCUSS WRITE: Hyperbolas in the Real World...Ch. 11.3 - DISCUSS: Light from a Lamp The light from a lamp...Ch. 11.4 - Suppose we want to graph an equation in x and y....Ch. 11.4 - The graphs of x2 = 12y and (x 3)2 = 12(y 1) are...Ch. 11.4 - The graphs of x252+y242=1 and (x3)252=(y1)242=1...Ch. 11.4 - The graphs of x242y232=1 and (x3)242(y1)232=1 are...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Ellipses An equation of an...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Parabolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Graphing Shifted Hyperbolas An equation of a...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Finding the Equation of a Shifted Conic Find an...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Complete the square to...Ch. 11.4 - Graphing Shifted Conics Use a graphing device to...Ch. 11.4 - Graphing Shifted Conics Use a graphing device to...Ch. 11.4 - Graphing Shifted Conics Use a graphing device to...Ch. 11.4 - Graphing Shifted Conics Use a graphing device to...Ch. 11.4 - Degenerate Conic Determine what the value of F...Ch. 11.4 - Common Focus and Vertex Find an equation for the...Ch. 11.4 - Confocal Parabolas This exercise deals with...Ch. 11.4 - Path of a Cannonball A cannon fires a cannonball...Ch. 11.4 - Orbit of a Satellite A satellite is in an...Ch. 11.4 - DISCUSS: A Family of Confocal Conics Conics that...Ch. 11.5 - Suppose the x- and y-axes are rotated through an...Ch. 11.5 - Consider the equation Ax2+Bxy+Cy2+Dx+Ey+F=0 (a) In...Ch. 11.5 - Rotation of Axes Determine the XY-coordinates of...Ch. 11.5 - Rotation of Axes Determine the XY-coordinates of...Ch. 11.5 - Rotation of Axes Determine the XY-coordinates of...Ch. 11.5 - Rotation of Axes Determine the XY-coordinates of...Ch. 11.5 - Rotation of Axes Determine the XY-coordinates of...Ch. 11.5 - Rotation of Axes Determine the XY-coordinates of...Ch. 11.5 - Finding the Equation for a Rotated Conic Determine...Ch. 11.5 - 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Graphing a Rotated Conic (a) Use the discriminant...Ch. 11.5 - Graphing a Rotated Conic (a) Use the discriminant...Ch. 11.5 - Graphing a Rotated Conic (a) Use the discriminant...Ch. 11.5 - Identifying a Conic from Its Discriminant (a) Use...Ch. 11.5 - Identifying a Conic from Its Discriminant (a) Use...Ch. 11.5 - Identifying a Conic from Its Discriminant (a) Use...Ch. 11.5 - Identifying a Conic from Its Discriminant (a) Use...Ch. 11.5 - Identifying a Hyperbola Using Rotation of Axes (a)...Ch. 11.5 - Identifying a Parabola Using Rotation of Axes (a)...Ch. 11.5 - Rotation of Axes Formulas Solve the equations...Ch. 11.5 - Graphing an Equation Using Rotation of Axes Show...Ch. 11.5 - PROVE: Matrix Form of Rotation of Axes Formulas...Ch. 11.5 - PROVE: Algebraic Invariants A quantity is...Ch. 11.5 - DISCOVER PROVE: Geometric Invariants Do you...Ch. 11.6 - All conics can be described geometrically by using...Ch. 11.6 - The polar equation of a conic with eccentricity e...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Finding a Polar Equation for a Conic Write a polar...Ch. 11.6 - Graphs of Polar Equations of Conics Match the...Ch. 11.6 - Graphs of Polar Equations of Conics Match the...Ch. 11.6 - Graphs of Polar Equations of Conics Match the...Ch. 11.6 - Graphs of Polar Equations of Conics Match the...Ch. 11.6 - Graphs of Polar Equations of Conics Match the...Ch. 11.6 - Graphs of Polar Equations of Conics Match the...Ch. 11.6 - Polar Equation for a Parabola A polar equation of...Ch. 11.6 - Polar Equation for a Parabola A polar equation of...Ch. 11.6 - Polar Equation for a Parabola A polar equation of...Ch. 11.6 - Polar Equation for a Parabola A polar equation of...Ch. 11.6 - Polar Equation for an Ellipse A polar equation of...Ch. 11.6 - Polar Equation for an Ellipse A polar equation of...Ch. 11.6 - Polar Equation for an Ellipse A polar equation of...Ch. 11.6 - Polar Equation for an Ellipse A polar equation of...Ch. 11.6 - Polar Equation for a Hyperbola A polar equation of...Ch. 11.6 - Polar Equation for a Hyperbola A polar equation of...Ch. 11.6 - Polar Equation for a Hyperbola A polar equation of...Ch. 11.6 - Polar Equation for a Hyperbola A polar equation of...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Identifying and Graphing a Conic (a) Find the...Ch. 11.6 - Rotating a Conic A polar equation of a conic is...Ch. 11.6 - Rotating a Conic A polar equation of a conic is...Ch. 11.6 - Rotating a Conic A polar equation of a conic is...Ch. 11.6 - Rotating a Conic A polar equation of a conic is...Ch. 11.6 - Families of Conics Graph the conics r = e/(1 e...Ch. 11.6 - Families of Conics (a) Graph the conics...Ch. 11.6 - Orbit of the Earth The polar equation of an...Ch. 11.6 - Perihelion and Aphelion The planets move around...Ch. 11.6 - Orbit of Pluto The distance from Pluto to the sun...Ch. 11.6 - DISCUSS: Distance to a Focus When we found polar...Ch. 11.6 - DISCUSS: Polar Equations of Orbits When a...Ch. 11 - (a) Give the geometric definition of a parabola....Ch. 11 - (a) Give the geometric definition of an ellipse....Ch. 11 - (a) Give the geometric definition of a hyperbola....Ch. 11 - (a) Suppose we are given an equation in x and y....Ch. 11 - (a) How can you tell whether the following...Ch. 11 - (a) Suppose that the x- and y-axes are rotated...Ch. 11 - (a) What is the discriminant of the equation in...Ch. 11 - (a) Write polar equations that represent a conic...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Parabolas An equation of a parabola is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Ellipses An equation of an ellipse is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Graphing Hyperbolas An equation of a hyperbola is...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Identifying and Graphing a Conic Determine whether...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Finding the Equation of a Conic Find an equation...Ch. 11 - Path of the Earth The path of the earth around the...Ch. 11 - LORAN A ship is located 40 mi from a straight...Ch. 11 - Families of Ellipses (a) Draw graphs of the...Ch. 11 - Families of Parabolas (a) Draw graphs of the...Ch. 11 - Identifying a Conic An equation of a conic is...Ch. 11 - Identifying a Conic An equation of a conic is...Ch. 11 - Identifying a Conic An equation of a conic is...Ch. 11 - Identifying a Conic An equation of a conic is...Ch. 11 - Identify a Conic from Its Graph Use a graphing...Ch. 11 - Identify a Conic from Its Graph Use a graphing...Ch. 11 - Identify a Conic from Its Graph Use a graphing...Ch. 11 - Identify a Conic from Its Graph Use a graphing...Ch. 11 - Polar Equations of Conics A polar equation of a...Ch. 11 - Polar Equations of Conics A polar equation of a...Ch. 11 - Polar Equations of Conics A polar equation of a...Ch. 11 - Polar Equations of Conics A polar equation of a...Ch. 11 - Find the focus and directrix of the parabola x2 =...Ch. 11 - Find the vertices, foci, and the lengths of the...Ch. 11 - Find the vertices, foci, and asymptotes of the...Ch. 11 - Find an equation for the parabola with vertex (0,...Ch. 11 - Find an equation for the ellipse with foci (3, 0)...Ch. 11 - Find an equation for the hyperbola with foci (0,...Ch. 11 - Find an equation for the conic whose graph is...Ch. 11 - Find an equation for the conic whose graph is...Ch. 11 - Find an equation for the conic whose graph is...Ch. 11 - Determine whether the equation represents an...Ch. 11 - Determine whether the equation represents an...Ch. 11 - Determine whether the equation represents an...Ch. 11 - Find an equation for the ellipse with center (2,...Ch. 11 - Find an equation for the parabola with focus (2,...Ch. 11 - A parabolic reflector for a car headlight forms a...Ch. 11 - (a) Use the discriminant to determine whether the...Ch. 11 - (a) Find the polar equation of the conic that has...Ch. 11 - Conics in Architecture The photographs on page 836...Ch. 11 - Constructing a Hyperbola In this problem we...Ch. 11 - A Parabola in a Rectangle The following method can...Ch. 11 - Hyperbolas from Straight Lines In this problem we...Ch. 11 - Tangent Lines to a Parabola In this problem we...Ch. 11 - 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