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mathematics
precalculus

Questions and Answers of Precalculus

Rotate the axes so that the new equation contains no xy-term. Analyze and graph the new equation.x2 + 4xy + y2 – 3 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 34x2 – 24xy + 41y2 – 25 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 25x2 – 36xy + 40y2 – 12√13x - 8√13y = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. x2 + 4xy + 4y2 + 5√5x + 5 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 4x2 – 4xy + y2 - 8√5x – 16√5y = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 11x2 + 10√3xy + y2 – 4 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 13x2 – 6√3xy + 7y2 – 16 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 3x2 – 10xy + 3y2 – 32 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. 5x2 + 6xy + 5y2 – 8 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. x2 – 4xy + y2 – 3 = 0
Determine the appropriate rotation formulas to use so that the new equation contains no xy-term. x2 + 4xy + y2 – 3 = 0
Identify the graph of each equation without completing the squares.2x2 + 2y2 – 8x + 8y = 0
Identify the graph of each equation without completing the squares.x2 + y2 – 8x + 4y = 0
Identify the graph of each equation without completing the squares.y2 – 8x2 – 2x – y = 0
Identify the graph of each equation without completing the squares.2y2 – x2 – y + x = 0
Identify the graph of each equation without completing the squares.4x2 – 3y2 – 8x + 6y + 1 = 0
Identify the graph of each equation without completing the squares.3x2 – 2y2 + 6x + 4 = 0
Identify the graph of each equation without completing the squares.2x2 + y2 – 8x + 4y + 2 = 0
Identify the graph of each equation without completing the squares.6x2 + 3y2 – 12x + 6y = 0
Identify the graph of each equation without completing the squares.2y2 – 3y + 3x = 0
Identify the graph of each equation without completing the squares.x2 + 4x + y + 3 = 0
True or False.To eliminate the xy-term from the equation x2 – 2xy + y2 – 2x + 3y + 5 = 0, rotate the axex through angle θ where cot θ = B2- 4AC.
True or False. The equation 3x2 + Bxy + 12y2 = 10 defines a parabola if B = -12.
True or False. The equation ax2 + 6y2 – 12y = 0 defines an ellipse a > 0.
Except for degenerate cases, the equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 defines ellipse if _________.
Except for degenerate cases, the equationAx2 + Bxy + Cy2 + Dx + Ey + F = 0, defines a(n) ______ if B2 – 4AC = 0.
To transform the equationAx2 + Bxy + Cy2 + Dx + Ey + F = 0, B ≠ 0into one in x' and y' without an x' y' term rotate the axes through an acute angle θ that satisfies the equation _____.
If θ is acute, the Half-angle Formula for the sine function is cosθ/2 = ______.
If θ is acute, the Half-angle Formula for the sine function is sinθ/2 = _____.
he Double-angle Formula for the sine function is sin2θ = ______.
The sum formula for the sine function is sin (A + B) = ______.
Show that the graph of an equation of the form Ax2 + Cy2 + Dx + Ey + F = 0, A ≠ 0, C ≠ 0,(a) Is a hyperbola if D2/4A + E2/4C – F ≠ 0.(b) Is two intersecting lines if D2/4A + E2/4C
Show that the graph of an equation of the formAx2 + Cy2 + F = 0, A ≠ 0, C ≠ 0, F ≠ 0where A and C are of opposite sign, is a hyperbola with center at (0, 0).
Prove that the hyperbola has the two oblique asymptotes y? = 1 У -x and y = - х
Two hyperbolas that have the same set of asymptotes are called conjugate. Show that the hyperbolas are conjugate. Graph each hyperbola on the same set of coordinate axes. x² 4 x? - y² = 1 and
A hyperbola for which is called an equilateral hyperbola. Find the eccentricity e of an equilateral hyperbola.
The eccentricity e of a hyperbola is defined as the number c/a where a is the distance of a vertex from the center and c is the distance of a focus from the center. Because c > a it follows that
Hyperbolas have interesting reflective properties that make them useful for lenses and mirrors. For example, if a ray of light strikes a convex hyperbolic mirror on a line that would (theoretically)
In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil 0.00004 cm thick.
Two recording devices are set 2400 feet apart, with the device at point A to the west of the device at point B. At a point between the devices, 300 feet from point B, a small amount of explosive is
Some nuclear power plants utilize “natural draft” cooling towers in the shape of a hyperboloid, a solid obtained by rotating a hyperbola about its conjugate axis. Suppose that such a cooling
Suppose that two people standing 1 mile apart both see a flash of lightning. After a period of time, the first person standing at point A hears the thunder. Two seconds later, the second person
Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time, the first person standing at point A hears the burst. One second later, the second
Analyze the conic.9x2 – y2 – 18x - 8y – 88 = 0
Analyze the conic.x2 – 6x – 8y - 31 = 0
Analyze the conic.x2 + 36y2 – 2x + 288y + 541 = 0
Analyze the conic.25x2 + 9y2 – 250x + 400 = 0
Analyze the conic.y2 = -12(x + 1)
Analyze the conic.x2 = 16(y – 3)
Analyze the conic. (y + 2)? (x – 2)² = 1 16 16 4
Analyze the conic. (x – 3)² y 25
Graph each function. Be sure to label any intercepts. f(x) = V-1 + x²
Graph each function. Be sure to label any intercepts. f(x) = -V-25 + x²
Graph each function. Be sure to label any intercepts. f(x) = -V9 + 9x²
Graph each function. Be sure to label any intercepts. f(x) = V16 + 4x²
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. x2 – 3y2 + 8x – 6y + 4 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. y2 – 4x2 – 16x – 2y – 19 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation.2x2 – x2 + 2x – 8y + 3 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. 4x2 – y2 – 24x – 4y + 16 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. 2x2 – y2 + 4x + 4y - 4 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. y2 – 4x2 - 4y - 8x - 4 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. y2 – x2 - 4y + 4x – 1 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. x2 – y2 - 2x - 2y – 1 = 0
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. (y – 3)2 – (x + 2)2 = 4
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. (x + 1)2 – (y + 2)2 = 4
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. (x + 4)2 – 9(y - 3)2 = 9
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. (y – 2)2 – 4(x + 2)2 = 4
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. (y + 3)² 4 (x – 2)2 = 1
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. (у + 3)? (х — 2)? 4 = 1
Find an equation for the hyperbola described. Graph the equation.Vertices at (1, -3) and (1, 1); asymptote the line 3 ) y + 1 =-(x - 1) 2
Find an equation for the hyperbola described. Graph the equation.Vertices at (-1, -1) and (3, -1); asymptote the line 3 |у +1 %3D— (х — 1)
Find an equation for the hyperbola described. Graph the equation.Center at (-4, 0); focus at (-4, 4); vertex at (-4, 2)
Find an equation for the hyperbola described. Graph the equation.Center at (3, 7); focus at (7, 7); vertex at (6, 7)
Find an equation for the hyperbola described. Graph the equation.Center at (1, 4); focus at (-2, 4); vertex at (0, 4)
Find an equation for the hyperbola described. Graph the equation.Center at (-3, -4); focus at (-3, -8); vertex at (-3, -2)
Find an equation for the hyperbola described. Graph the equation.Center at (-3, 1); focus at (-3, 6); vertex at (-3, 4)
Find an equation for the hyperbola described. Graph the equation.Center at (4, -1); focus at (7, -1); vertex at (6, -1)
Write an equation for each hyperbola. y = 2 x y = -2 x YA 5- -5 5х -5
Write an equation for each hyperbola. Ул y = 2 x y = -2 x 10 5 х -5 -10F
Write an equation for each hyperbola. y = x 3 3 x -3 -3F
Write an equation for each hyperbola. Уд 3 У —х -3 3х
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. 2x2 – y2 = 4
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. y2 – x2 = 25
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. x2 – y2 = 4
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. y2 – 9x2 = 9
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. 4y2 – x2 = 16
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. 4x2 – y2 = 16
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. x2/16 – y2/4 = 1
Find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. x2/25 – y2/9 = 1
Find an equation for the hyperbola described. Graph the equation. Foci at (0, -2) and (0, 2); asymptote the line y = -x
Find an equation for the hyperbola described. Graph the equation. Foci at (-4, 0) and (4, 0); asymptote the lin y = -x
Find an equation for the hyperbola described. Graph the equation. Vertices at (-4, 0); and (4, 0); asymtote line y = 2x
Find an equation for the hyperbola described. Graph the equation. Vertices at (0, -6); and (0, 6); asymtote line y = 2x
Find an equation for the hyperbola described. Graph the equation. Focus at (0, 6); vetices at (0, -2) and (0, 2)
Find an equation for the hyperbola described. Graph the equation. Foci at (-5, 0); focus at (5, 0); vertex at (3, 0)
Find an equation for the hyperbola described. Graph the equation. Center at (0, 0); focus at (-3, 0); vertex at (2, 0)
Find an equation for the hyperbola described. Graph the equation. Center at (0, 0); focus at (0, -6); vertex at (0, 4)
Find an equation for the hyperbola described. Graph the equation. Center at (0, 0); focus at (0, 5); vertex at (0, 3)
Find an equation for the hyperbola described. Graph the equation. Center at (0, 0); focus at (3, 0); vertex at (1, 0)
The graph of a hyperbola is given. Match each graph to its equation.(A) x2 – y2 = 1(B) x2 – y2/4 = 1(C) y2/4 – x2 = 1(D) y2 – x2/4 = 1 Ул 3 -3 3х -3F

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