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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

For the following exercises, convert the polar equation of a conic section to a rectangular equation. r = 6sec 0 -2 +3 sec 0
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.x2 + 4x + 8y − 4 = 0
For the following exercises, determine the angle θ that will eliminate the xy term and write the corresponding equation without the xy term.4x2 − 2√3 xy + 6y2 − 1 = 0
For the following exercises, find the equations of the asymptotes for each hyperbola.16y2 + 96y − 4x2 + 16x + 112 = 0
For the following exercises, find the foci for the given ellipses.x2 + 4y2 + 4x + 8y = 1
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r= 6csc 0 3+2 csc 0
For the following exercises, write the equation of the parabola using the given information. Focus at (−4,0); directrix is x = 4
For the following exercises, graph the parabola, labeling the focus and the directrix. x = 1/8 y2
For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation. y = − x2, θ = − 45°
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci. x2/49 − y2/16 = 1
For the following exercises, find the foci for the given ellipses.10x2+ y2 + 200x = 0
For the following exercises, write the equation of the parabola using the given information.Focus at (2, 9/8) ; directrix is y = 7/8
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label
For the following exercises, graph the parabola, labeling the focus and the directrix.y = 36x2
For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.x = y2, θ = 45°
For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.x2/64 − y2/4 = 1
For the following exercises, graph the given ellipses, noting center, vertices, and foci. x2/25 + y2/36 = 1
For the following exercises, write the equation of the parabola using the given information.A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver,
For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label
For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation. 4 + 1 = 1,0 = 45°
For the following exercises, use Cramer’s Rule to solve the linear systems of equations. 3 10 x-y- 3 10 1/x-1/09 - 11/12 = ²x-1-y-z=-1²-2 1 50 9 50
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.4x2 − 8x − 9y2 −
For the following exercises, determine which conic section is represented by the given equation, and then determine the angle θ that will eliminate the xy term.x2 + 4xy + 4y2 + 6x − 8y = 0
For the following exercises, graph the hyperbola, labeling vertices and foci.2y2 − x2 − 12y − 6 = 0
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.4x2 − 8x + 9y2 − 72y + 112 = 0
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r = 2 53 sin 0
For the following exercises, find a new representation of the given equation after rotating through the given angle.4x2 − xy + 4y2 − 2 = 0, θ = 45°
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.(y−4)2 = 2(x+3)
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.−9x2 − 54x + 9y2 −
For the following exercises, rewrite in the x'y' system without the x'y' term, and graph the rotated graph. 11x2 + 10√3 xy + y2 = 4
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.9x2 − 54x + 9y2 − 54y + 81 = 0
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r = 8 3 - 2 cos 0
For the following exercises, find the equation of the hyperbola. Center at (0,0), vertex at (0,4), focus at (0,−6)
For the following exercises, find a new representation of the given equation after rotating through the given angle.2x2 + 8xy − 1 = 0, θ = 30°
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.4x2 − 24x − 36y2 −
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.(x+1)2 = 2(y+4)
For the following exercises, rewrite in the x'y' system without the x'y' term, and graph the rotated graph.16x2 + 24xy + 9y2 − 125x = 0
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.4x2 − 24x + 36y2 − 360y + 864 = 0
For the following exercises, find the equation of the hyperbola.Foci at (3,7) and (7,7), vertex at (6,7)
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r= 3 2 + 5 cos 0
For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity. r= 2 3 sin 0
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.−4x2 + 24x + 16y2 −
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.(x+4)2 = 24(y+1)
For the following exercises, find a new representation of the given equation after rotating through the given angle.−2x2 + 8xy + 1 = 0, θ = 45°
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.4x2 + 24x + 16y2 − 128y + 228 = 0
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r 4 2 + 2 sin 0
For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix. y2 = 12x
For the following exercises, find a new representation of the given equation after rotating through the given angle.4x2 +√2 xy + 4y2 + y + 2 = 0, θ = 45°
For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity. r= 5 4 + 6 cos 0
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.−4x2 + 40x + 25y2 −
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.(y+4)2 = 16(x+4)
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.4x2 + 40x + 25y2 − 100y + 100 = 0
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r= 3 88 cos 0
For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.(x + 2)2 = 1/2 (y − 1)
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.x2 + 2x − 100y2 −
For the following exercises, determine the angle θ that will eliminate the xy term and write the corresponding equation without the xy term. x2 + 3√3 xy + 4y2 + y − 2 = 0
For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci. r= 12 4-8 sin 0
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.y2 + 12x − 6y + 21 = 0
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r= 2 6 + 7 cos 0
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.x2 + 2x + 100y2 − 1000y + 2401 = 0
For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.y2 − 6y − 6x − 3 = 0
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.−9x2 + 72x + 16y2 +
For the following exercises, determine the angle θ that will eliminate the xy term and write the corresponding equation without the xy term.4x2 + 2√3 xy + 6y2 + y − 2 = 0
For the following exercises, graph the given conic section. If it is a parabola, label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci. r = 2 4 + 4 sin 0
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.4x2 + 24x + 25y2 + 200y + 336 = 0
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r = 5 5-11 sin 0
For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix.x2 + 10x − y + 23 = 0
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.(y−2)2 = 4/5 (x+4)
For the following exercises, convert the polar equation of a conic section to a rectangular equation. r 4 1+ 3 sin 0
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. (x-7)² 49 + (y-7)² 49 = 1
For the following exercises, find a new representation of the given equation after rotating through the given angle. 3x2 + xy + 3y2 − 5 = 0, θ = 45°
For the following exercises, graph the hyperbola, labeling vertices and foci.x2 − 4y2 + 6x + 32y − 91 = 0
For the following exercises, determine which conic section is represented by the given equation, and then determine the angle θ that will eliminate the xy term. 3x2 − 2xy + 3y2 = 4
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. (x - 2)² 49 (y + 7)²
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.(x − 1)2 = 4(y − 1)
For the following exercises, determine which conic section is represented based on the given equation.8x2 + 4√2 xy + 4y2 − 10x + 1 = 0
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.r(7 + 8cos θ) = 7
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. (x + 5)² 4 + (y-7)² 9 = 1
For the following exercises, graph the hyperbola, labeling vertices and foci. (y-1)² (x + 1)² 49 4 = 1
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. (y-6)² 36 (x +
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.x = 1/36 y2
For the following exercises, determine which conic section is represented based on the given equation.−x2 + 4√2 xy + 2y2 − 2y + 1 = 0
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.r(4 − 5sin θ) = 1
For the following exercises, graph the hyperbola, labeling vertices and foci. x2/9 − y2/16 = 1
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.Write the equation of a parabola with a focus at (2,3) and directrix y = −1.
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes. (x-1)²(y-2)² = 1 9 16
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.x = 36y2
For the following exercises, determine which conic section is represented based on the given equation.2x2 + 4√3 xy + 6y2 − 6x − 3 = 0
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci. (x - 2)² 49 + (y - 4)² 25 = 1
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.r(3 + 5sin θ) = 11
For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.3x2 − y2 − 12x − 6y − 9 = 0
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.y2 + 8x − 8y + 40 = 0
For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.9y2 − 4x2 = 1
For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola.x = 1/8 y2
For the following exercises, determine which conic section is represented based on the given equation.−3x2 + 3√3 xy − 4y2 + 9 = 0
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.r(1 − cos θ) = 3
For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.4x2 + 16y2 = 1
For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci.9y2 − 4x2 + 54y − 16x + 29 = 0
For the following exercises, graph the parabola, labeling the vertex, focus, and directrix.(x − 1)2 = −4(y + 3)

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