Algebra and Trigonometry (10th Edition)

Chapter 12

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; The distance  is the length of the hypotenuse... more

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4 ; To complete the square, always take half of ... more

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and ; Take the square root of both sides. ; Make ... more

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; has -axis symmetry to .  Change the sign of -... more

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3 ; shifts  right  units and up  units. ; up ; ... more

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parabola ; This is the definition of parabola.  ... more

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axis of symmetry ; This is the definition of axis ... more

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latus rectum ; This is the definition of latus ... more

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The -term is squared so the axis of symmetry is ... more

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; The vertex is the intersection of the axis of ... more

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The -coordinate of the focus must be the same as ... more

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This is a vertical line to the left of the vertex ... more

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; ; Since the focus is to the right of the vertex... more

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;; ; Since the focus is above the vertex, the ... more

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;; ; Since the focus is to the left of the vertex... more

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;; ; Since the focus is to the left of the vertex... more

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;; ; Since the focus is to the left of the ... more

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;;  ; Since the focus is below the directrix, the ... more

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;; ; The directrix is horizontal so the axis of ... more

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;;  ; The directrix is vertical so the axis of ... more

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;; ; Since the axis of symmetry is vertical and ... more

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; ; ; Since the axis of symmetry is horizontal and... more

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;;  ; Since the focus is below the vertex, the ... more

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;; ; Since the focus is to the right of the vertex... more

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;; ; Since the focus is to the right of the vertex... more

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;; ; Since the focus is below the vertex, the ... more

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;; ; Since the focus is above the directrix, the ... more

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; ;  ; Since the focus is to the right of the ... more

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; ;  ; Since the focus is to the left of the ... more

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; ; ; Since the focus is above the directrix, the ... more

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; ; ; Since the term is squared and the ... more

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; ;  ; Since the term is squared and the ... more

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; ;  ; Since the term is squared and the ... more

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; ; ; Since the term is squared and the ... more

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; ; ; Since the term is squared and the ... more

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; ; ; Since the term is squared and the ... more

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; ;  ; Since the term is squared and the ... more

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; ;   ; Since the  term is squared and the ... more

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; ; ; Since the term is squared and the ... more

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; ;  ; To complete the square, add the square of ... more

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; ; ; To complete the square, add the square of ... more

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; ; ; To complete the square, add the square of ... more

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; ;  ; To complete the square, add the square of ... more

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; ;   ; To complete the square, add the square of ... more

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; ; ; To complete the square, add the square of ... more

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; ; ; To complete the square, add the square of ... more

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;; ; To complete the square, add the square of ... more

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; The parabola opens right and has equation .  ... more

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; The parabola opens down and has equation .  The... more

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; The parabola opens left and has equation .  The... more

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; The parabola opens up and has equation .  The ... more

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;  The parabola opens up and has equation .  The ... more

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; The parabola opens up and has equation .  The ... more

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;  The parabola opens right and has equation .  ... more

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;  The parabola opens left and has equation .  ... more

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1.5625 ft directly above the vertex        ; The ... more

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; The parabola opens up, has the vertex at the ... more

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1 inch directly above vertex ; The parabola opens ... more

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5.65686 inches ; The parabola opens up, has the ... more

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20 ft ; The parabola opens up, has the vertex at ... more

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15.625 ft ; The parabola opens up, has the vertex ... more

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0.78125 ft ; The parabola opens up,  the vertex is... more

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11.3138 ft ; The parabola opens up,  the vertex is... more

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4.17 ft directly above the vertex ; The parabola ... more

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0.33 ft directly above the vertex ; The parabola ... more

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24.31 ft; 18.75 ft;7.64 ft ; If the vertex is the ... more

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15.63 ft ; If the vertex is the highest point and ... more

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The equation  is in the form  where .  Therefore, ... more

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The equation  is in the form  where .  Therefore, ... more

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;;;-axis symmetry ; To find -intercept substitute... more

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; Since 4 and 8 are powers of 2, substitute  and... more

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; ; ; ; ; Apply the Pythagorean identity .Solve ... more

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; Let be .Apply the property of inverse ... more

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; Use the distance formula  with  as  and  as . ... more

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; To find the -intercepts substitute into and ... more

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; The point  is symmetric with respect to the -... more

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left 1; down 4 ; Substituting and into yields... more

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; The circle  has center  and center .  ... more

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ellipse ; The two fixed points are the foci.  A ... more

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The minor axis in an ellipse is the line through ... more

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; This equation of this ellipse is in the form of... more

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5 ; This equation of this ellipse is in the form ... more

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; Apply (h, k) to the center (2, -3) and and .  ... more

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Since the -coordinates of the foci are the same, ... more

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Center: ; Vertices:,; Foci: ,  ; The graph is an ... more

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Center: ; Vertices, ;Foci: , ; The graph is an ... more

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Center: ; Vertices: ,;Foci: ,  ; The graph is an ... more

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Center: ; Vertices: , ;Foci ,   ; The graph is an ... more

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Center: ; Vertices, ;Foci , ; The standard form of... more

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Center: ; Vertices: ,; Foci , ; The standard form ... more

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Center: ; Vertices, ; Foci , ; The standard form ... more

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Center: ;Vertices ,;Foci , ; The standard form of ... more

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Center: ; Vertices , , ,Focus: ; The standard form... more

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Center: ; Vertices: , , , ; Focus: ; The standard ... more

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; Since the -coordinates of the focus and vertex ... more

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; Since the -coordinates of the focus and vertex ... more

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; Since the -coordinates of the focus and vertex ... more

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; Since the -coordinates of the focus and vertex ... more

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; Since the foci are on the -axis and each focus ... more

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; Since the foci are on the -axis and each focus ... more

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; Since the vertices are on the -axis and each ... more

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; Since the vertices are on the -axis and each ... more

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; Since the foci are on the -axis and each focus ... more

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; Since the vertices are on the -axis and each ... more

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; Since the vertex is on the -axis and the center... more

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; Since the vertices are on the -axis and each ... more

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; Applying to the center  yields and . The ... more

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; Applying to the center  yields and .  The ... more

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; Applying to the center  yields and .  The ... more

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; Applying to the center  yields and . The ... more

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Center: ; Foci: ,; Vertices:; ; Since the term is... more

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Center: ; Foci: ,;Vertices: , ; Since the term is... more

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Center: ; Foci: ,; Vertices:, ; The standard form ... more

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Center: ; Foci: , ; Vertices: ; ; The standard ... more

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Center: ; Foci: ,; Vertices, ; Since the ... more

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Center: ,Foci: , ; Vertices: ; Write the equation ... more

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Center: ; Foci: , ; Vertices: ; ; Write the ... more

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Center: ; Foci: ,; Vertices , ; Write the equation... more

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Center: ; Foci: ,; Vertices: , ; Write the ... more

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Center: ; Foci: ,; Vertices: , ; Write the ... more

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Center: ; Foci: ,; Vertices, ; Write the equation ... more

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Center: ; Foci: ,; Vertices , ; Rearrange the ... more

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; Since the -coordinates of the focus and vertex ... more

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; Since the -coordinates of the focus and vertex ... more

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; The center is the midpoint of the vertices. Use... more

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; The center is the midpoint of the foci. Use the... more

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; The center is the midpoint of the foci. Use the... more

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; The center is the midpoint of the vertices. Use... more

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; Since the -coordinates of the center and focus ... more

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; Since the -coordinates of the center and focus ... more

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; Since the -coordinates of the vertex and the ... more

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; Since the -coordinates of the vertex and the ... more

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To find the -intercept, replace the left hand side... more

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To find the -intercept, replace the left hand side... more

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To find the -intercept, replace the left hand side... more

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To find the -intercept, replace the left hand side... more

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; If the center is placed at the origin, then the... more

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12 ft; 4.55 ft; 2.57 ft; 2 ft ; If the center is ... more

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43.3 ft ; If the center is placed at the origin, ... more

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ft, ft ; If the center is placed at the origin, ... more

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24.65 ft; 21.65 ft; 13.82 ft ; If the center is ... more

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ft ; If the center is placed at the origin, then ... more

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30 ft ; If the center is placed at the origin, ... more

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73.69 ft ; If the center is placed at the origin, ... more

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It needs to be an ellipse with major axis of in ... more

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555.9731 cubic inches ; The length of the object ... more

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; ; as it is the distance from the center to the ... more

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; ; as it is the distance from the center to the ... more

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; ; ; as it is the distance from the center to the... more

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;; ; as it is the distance from the center to the ... more

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35 million miles ; because it is the distance from... more

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28 sq ft ; is half of the length of the major axis... more

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; The conic section equation can be put into the ... more

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; ; ; ; To find the zeros of the function, set the... more

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; ; , no oblique asymptote. ; The domain of a ... more

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ft-lb ; Substitutefor andfor into the formula ... more

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; ; ; Substitute 14 for and for into the formula... more

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; The distance formula is .  Let and .   ; more

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; The leading coefficient is 1.  To complete the ... more

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; ; To find the -intercepts, set and solve the ... more

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The equation has -axis symmetry since is ... more

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right 5; down 4 ; The equation in the from is the... more

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;; ; The domain of a rational function can be ... more

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hyperbola ; For a hyperbola, the difference of the... more

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transverse axis ; The transverse axis of a ... more

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This is a hyperbola. However, because the term is... more

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; ; Apply center to . Because the hyperbola has ... more

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; ; Apply center to .  Because the hyperbola has ... more

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When in standard from, the equation for a ... more

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2 ; The equation is in the form , which is a ... more

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; ; The equation is in the form which is a ... more

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; Since the center is the origin with the focus ... more

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; Since the center is the origin with the focus ... more

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; Since the center is the origin with the focus ... more

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; Since the center is the origin with the focus ... more

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; The center is the midpoint of the foci.  ... more

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; The center is the midpoint of the vertices.  ... more

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; The center is the midpoint of the vertices.  ... more

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; The center is the midpoint of the vertices.  ... more

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; The center is the midpoint of the foci.  ... more

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; The center is the midpoint of the foci.  ... more

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Center ; transverse axis is -axis; vertices and... more

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Center ; transverse axis is -axis; vertices and... more

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Center ; transverse axis is -axis; vertices and ;... more

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Center ; transverse axis is -axis; vertices and ;... more

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Center ; transverse axis is -axis; vertices and ;... more

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Center ; transverse axis is -axis; vertices and... more

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Center ; transverse axis is -axis; vertices and... more

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Center ; transverse axis is -axis; vertices and... more

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; The hyperbola has the origin as its center and ... more

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; The hyperbola has the origin as its center and ... more

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; The hyperbola has the origin as its center and ... more

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; The hyperbola has the origin as its center and ... more

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; Since the center, focus, and vertex all have ... more

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; Since the center, focus, and vertex all have ... more

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; Since the center, focus, and vertex all have ... more

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; Since the center, focus, and vertex all have ... more

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; The center is the midpoint of the foci.  ... more

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; The center is the midpoint of the vertices.  ... more

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; The center is the midpoint of the vertices.  ... more

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; The center is the midpoint of the vertices.  ... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Find the -intercept(s) by setting the function ... more

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Find the -intercept(s) by setting the function ... more

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Find the -intercept(s) by setting the function ... more

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Find the -intercept(s) by setting the function ... more

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Center ; transverse axis is parallel to the -axis... more

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Center ; transverse axis is parallel to the -axis... more

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Vertex: ; Focus ;   Directrix ; Since the term is... more

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Vertex ; Focus: ; Directrix ; Since the term is ... more

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Center ; Foci , ; Vertices , ; To complete the ... more

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Center: Foci: ,, Vertices , ; To complete the ... more

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Vertex ; Focus ; Directrix ; To complete the ... more

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Center ; transverse axis is parallel to the -axis... more

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The noise event occurs 50,138 feet to the north of... more

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The noise event occurs 5236 feet to the north of ... more

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; Sketch the hyperbola that generated the ... more

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; If the time difference for both explosions is ... more

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; Compare the hyperbolic equation with .  So and... more

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When is close to 1, the branches of the hyperbola... more

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; Find .  Find the eccentricity.  ; more

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The hyperbolas are conjugate and the asymptotes ... more

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Solve the equation of the hyperbola for . As , the... more

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Rewrite the equation: Since and are opposite in ... more

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Amplitude: ; Period: ; Phase shift: ; Compare the ... more

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; ; ; Substitute 7 for , 10 for , and for into ... more

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Circle with radius 3 and center at ; Multiply both... more

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Circle with radius 3 and center at ; Multiply both... more

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; The sum formula of the sine function states ... more

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; The double-angle formula for sine function is ... more

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; Use the double angle formula of .Replace  by  ... more

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; Use the double angle formula of .Replace by ... more

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; In order to transform the given equation into a... more

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Parabola ; The equation , where defines a ... more

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; The equation , where defines an ellipse if ... more

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Compare the general second degree polynomial ... more

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Compare the general second degree polynomial ... more

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To transform the equation , where  into an ... more

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Parabola ; Compare the given equation with . It ... more

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Parabola ; Compare the given conic equation with... more

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Ellipse ; Compare the given equation with . It ... more

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Ellipse ; Compare the given equation with . It ... more

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Hyperbola ; Compare the given equation with . It ... more

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Hyperbola ; Compare the given equation with . It ... more

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Hyperbola ; Compare the given equation with . It ... more

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Hyperbola ; Compare the given equation with . It ... more

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Circle ; Compare the given equation with . It ... more

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Circle ; Compare the given equation with . It ... more

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;  ; Compare the given equation with . It gives... more

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;  ; Compare the given equation with . It gives... more

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;  ; Compare the given equation with . It gives... more

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;  ; Compare the given equation with . It gives... more

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;  ; Comparison of the given equation with . It ... more

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;  ; Compare the given equation with . It gives... more

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; ; Compare the given equation with . It gives... more

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; ; Compare the given conic equation with , which ... more

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; ; Compare the given conic equation with that ... more

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; ; Compare the given conic equation with  that ... more

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Hyperbola with center at , vertices at , and -axis... more

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Hyperbola with center at , vertices are at and - ... more

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Ellipse with center at  and vertices at  on -axis ... more

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Hyperbola with center at , vertices at , and - ... more

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Ellipse with center at , vertices are at and - ... more

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Hyperbola with center at, vertices are , and - ... more

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Parabola with vertex at , focus at ,and - axis is ... more

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Parabola with vertex at , focus at , and axis of ... more

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Ellipse with center at , vertices are and , and ... more

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Ellipse with center at , vertices and the major ... more

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Parabola with vertex , focus at and the axis of ... more

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Parabola with vertex at origin, focus at , and -... more

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Hyperbola ; On comparing the given conic equation ... more

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Ellipse ; Compare the given equation with general ... more

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Hyperbola ; Compare the given equation with ... more

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Ellipse ; Compare the given equation with general ... more

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Parabola ; Compare the given equation with general... more

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Ellipse ; Compare the given equation with general ... more

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Ellipse ; Compare the given equation with general ... more

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Parabola ; Compare the given equation with general... more

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Ellipse ; Compare the given equation with general ... more

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Ellipse ; Compare the given equation with general ... more

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; ; ; ; ; ; Apply the rotation formula on the ... more

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Entire Proof Answer: Apply the rotation formula ... more

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Entire Proof Answer: Apply the rotation formula ... more

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Entire Proof Answer: Use the rotation formula and ... more

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Entire Proof Answer: Rewrite the given equation in... more

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The equation, , defines a parabola if , an ellipse... more

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The given equation defines a parabola if , an ... more

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; ; ; Draw a triangle with the help of given ... more

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; Sketch a triangle with the help of given ... more

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; Substitute  for  and  for  into the given ... more

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; Compare the given complex number with the ... more

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; In the polar coordinate system, a point can be ... more

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; Multiply both sides of the given equation by... more

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conic ; The conic section is defined as the set of... more

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1 ; The ratio of the distance between the focus ... more

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Rewrite the given equation in standard form.... more

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The standard equation of an ellipse with center ... more

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ParabolaDirectrix is perpendicular to the polar ... more

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ParabolaDirectrix is parallel to the polar axis at... more

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HyperbolaDirectrix is parallel to the polar axis ... more

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HyperbolaDirectrix is perpendicular to the polar ... more

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EllipseDirectrix is perpendicular to the polar ... more

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EllipseDirectrix is parallel to the polar axis at ... more

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ParabolaDirectrix of the parabola is perpendicular... more

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ParabolaDirectrix is parallel to the polar axis, 3... more

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EllipseDirectrix is parallel to the polar axis and... more

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EllipseVertices: , Directrix is perpendicular to ... more

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HyperbolaVertices: and Directrix is perpendicular... more

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HyperbolaVertices: , Directrix is parallel to the... more

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EllipseDirectrix is parallel to the polar axis and... more

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HyperbolaVertices: and Directrix is perpendicular... more

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EllipseDirectrix is parallel to the polar axis and... more

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EllipseDirectrix is perpendicular to the polar ... more

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EllipseDirectrix is perpendicular to the polar ... more

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ParabolaDirectrix of the parabola is parallel to ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Rearrange the given equation. ; Square both ... more

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; Write the given equation in terms of . ; ... more

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; Write the given equation in terms of . ; ... more

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; Write the polar equation for the directrix ... more

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; Write the polar equation for the directrix ... more

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; Write the polar equation for the directrix ... more

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; Write the polar equation for the directrix ... more

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; Write the polar equation for the directrix ... more

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; Write the polar equation for the directrix ... more

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Entire Proof Answer:The directrix D is on the ... more

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Entire Proof Answer: The directrix D is above the ... more

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Entire Proof Answer: The directrix D is below the ... more

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Greatest: milesShortest: miles ; For the ... more

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; Substitute  for ,  for , and  for  into the ... more

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Amplitude: 4; Period: ; Compare the given equation... more

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; Factor the trinomial on the left side of the ... more

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26 ; Apply the formula for the magnitude of a ... more

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3 ; Compare the given expression with , where is ... more

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plane ; Each graph is the collection of the points... more

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Ellipse ; The given parametric have sine and ... more

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Cycloid ; Assume a circle of radius, , which rolls... more

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Assume two parametric equations.Solve the ... more

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Parametric equations are used to describe the ... more

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; Take some random values of in the interval ... more

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; Take some random values of in the interval ... more

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; Take some random values of in the interval ... more

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; Take some random values of in the interval ... more

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; Take some random values of in the interval ... more

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