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mathematics
precalculus 1st

Questions and Answers of Precalculus 1st

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.y2/4 − x2/81 = 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.y = −4x2
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. r= 4 7 + 2 cos 0
For the following exercises, determine which conic section is represented based on the given equation.3x2 + 6xy + 3y2 − 36y − 125 = 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 + 9y2 = 1
For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci. (y + 1)² 16 (x-4)² 36 = 1
For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix.3x2 − 12x − y + 11 = 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.x2/100 − y2/9 = 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.y = 1/4 x2
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. r= 2 1 - cos 0
For the following exercises, determine which conic section is represented based on the given equation.4x2 + 9xy + 4y2 − 36y − 125 = 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.x2/100 + y2/64 =1
For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci. x2/81 − y2/9 = 1
For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix. y2 + 10x = 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. x = 8y2
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. x2/25 − y2/36 = 1
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. r = 3 10+ 10 cos 0
For the following exercises, determine which conic section is represented based on the given equation.2x2 + 3y2 − 8x − 12y + 2 = 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. x2/4 + y2/49 =1
For the following exercises, use the given information to find the equation for the ellipse.A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the
For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.Write the standard form equation of a hyperbola with foci at (1,0) and
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.y2 + 12x − 6y − 51 = 0
For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.−9x2 + 18x + y2 + 4y − 14 = 0
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. r = 16 4+3 cos 0
For the following exercises, determine which conic section is represented based on the given equation.4y2 − 5x + 9y + 1 = 0
For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.4x2 − 8x + 9y2 − 72y + 112 = 0
For the following exercises, use the given information to find the equation for the ellipse.Center at (2,−2), vertex at (7,−2), focus at (4,−2)
For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes.y2 − x2 + 4y − 4x − 18 = 0
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.(y − 3)2 = 8(x − 2)
For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.25x2 − 16y2 = 400
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. r= 5 1 + 2 sin 0
For the following exercises, determine which conic section is represented based on the given equation.4x2 − y2 + 8x − 1 = 0
For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.4x2 + 9y2 = 1
For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes. (x - 3)² 25 (y + 3)² 1 = 1
For the following exercises, use the given information to find the equation for the ellipse. Center at (0,0), focus at (3, 0), vertex at (−5,0)
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.3x2 − 6y2 = 12
For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.5y2 + 4x2 = 6x
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. r= 8 4- 3 cos 0
For the following exercises, determine which conic section is represented based on the given equation.2x2 − 2y2 + 4x − 6y − 2 = 0
For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.4x2 − y2 = 4
For the following exercises, graph the ellipse, noting center, vertices, and foci.2x2 + 3y2 − 20x + 12y + 38 = 0
For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes.16y2 − 9x2 + 128y + 112 = 0
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form.y = 4x2
For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form.x2/36 − y2/9 = 1
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity. r= 3 44 sin 0
For the following exercises, determine which conic section is represented based on the given equation.x2 − 10x + 4y − 10 = 0
For the following exercises, graph the ellipse, noting center, vertices, and foci.4x2 + y2 + 16x + 4y − 44 = 0
For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form.4x2 + 9y2 = 36
For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes. x2 49 81 = 1
For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. y2 = 4 − x2
For the following exercises, determine which conic section is represented based on the given equation. 9x2 + 4y2 + 72x + 36y − 500 = 0
For the following exercises, determine whether the following equations represent hyperbolas. If so, write in standard form. 3y2 + 2x = 6
For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.r = 6/1 − 2 cos θ
For the following exercises, determine whether the given equations represent ellipses. If yes, write in standard form. 2x2 + y = 4
For the following exercises, graph the ellipse, noting center, vertices, and foci.(x − 4)2/25 + (y + 3)2/49 = 1
For the following exercises, sketch the graph, identifying the center, vertices, and foci.A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet
As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?
For the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the value of θ that satisfies cot(2θ) = A−C/B gives us what information?
Where must the center of hyperbola be relative to its foci?
What do we know about the focus/foci of a conic section if it is written as a polar equation?
What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y-axis?
For the following exercises, graph the ellipse, noting center, vertices, and foci. x2/36 + y2/9 = 1
For the following exercises, sketch the graph, identifying the center, vertices, and foci.Write the standard form equation of an ellipse with a center at (1,2), vertex at (7,2), and focus at (4,2).
For the following exercises, sketch the graph, identifying the center, vertices, and foci.2x2 + y2 + 8x − 6y − 7 = 0
What is the effect on the graph of a parabola if its equation in standard form has increasing values of p?
If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?
If the transverse axis of a hyperbola is vertical, what do we know about the graph?
Given the equation ax2 + 4x + 3y2 − 12 = 0,what can we conclude if a > 0?
For the special case mentioned in the previous question, what would be true about the foci of that ellipse?
For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.9x2 + 36y2 − 36x + 72y + 36 = 0
For the following exercises, sketch the graph, identifying the center, vertices, and foci. (x - 3)² (y-2)² + 64 36 1
If the equation of a parabola is written in standard form and p is negative and the directrix is a horizontal line, then what can we conclude about its graph?
If a conic section is written as a polar equation, and the denominator involves sin θ, what conclusion can be drawn about the directrix?
What must be true of the foci of a hyperbola?
If the equation of a conic section is written in the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, and B2 − 4AC > 0, what can we conclude?
What special case of the ellipse do we have when the major and minor axis are of the same length?
For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.9x2 + y2 + 54x − 4y + 76 = 0
For the following exercises, write the equation in standard form and state the center, vertices, and foci.9y2 + 16x2 − 36y + 32x − 92 = 0
If the equation of a parabola is written in standard form and p is positive and the directrix is a vertical line, then what can we conclude about its graph?
If a conic section is written as a polar equation, what must be true of the denominator?
What can we conclude about a hyperbola if its asymptotes intersect at the origin?
If the equation of a conic section is written in the form Ax2 + By2 + Cx + Dy + E = 0 and AB = 0, what can we conclude?
Where must the foci of an ellipse lie?
For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.(x − 2)2/100 + (y + 3)2/36 = 1
For the following exercises, write the equation in standard form and state the center, vertices, and foci. x2/9 + y2/4 = 1
Define a parabola in terms of its focus and directrix.
Define a hyperbola in terms of its foci.
What effect does the xy term have on the graph of a conic section?
Define an ellipse in terms of its foci.
For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci.x2/25 + y2/64 = 1
For the following exercises, use Cramer’s Rule to solve the linear systems of equations. 4x - 3y + 5z = - 7x9y3z= x-5y5z || 3 2 5 2 5 2
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.x + 6y + 3z = 42x + y + 2z = 33x − 2y + z = 0
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.−0.5x + 0.1y = 0.3−0.25x + 0.05y = 0.15
For the following exercises, use a system of linear equations with two variables and two equations to solve.Admission into an amusement park for 4 children and 2 adults is $116.90. For 6 children and
For the following exercises, use Cramer’s Rule to solve the linear systems of equations.0.2x − 0.1y = 0−0.3x + 0.3y = 2.5
For the following exercises, use a system of linear equations with two variables and two equations to solve.A concert manager counted 350 ticket receipts the day after a concert. The price for a
For the following exercises, solve the system for x, y, and z.At one time, in the United States, 398 species of animals were on the endangered species list. The top groups were mammals, birds, and
For the following exercises, write a system of equations to solve each problem. Solve the system of equations. Students were asked to bring their favorite fruit to class. 90% of the fruits
For the following exercises, use a system of linear equations with two variables and two equations to solve.If a scientist mixed 10% saline solution with 60% saline solution to get 25 gallons of 40%
For the following exercises, use this scenario: A health-conscious company decides to make a trail mix out of almonds, dried cranberries, and chocolate-covered cashews. The nutritional information

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