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mathematics
precalculus

Questions and Answers of Precalculus

The parametric equations x(t) = 2 sin t y(t) = 3 cos t define a(n) ____________.(a) Circle(b) Ellipse(c) Hyperbola(d) Parabola
A___________ is the set of points P in a plane for which the ratio of the distance from a fixed point F, called the_________, to P to the distance from a fixed line D, called the__________, to P
Except for degenerate cases, the equationdefines a(n)______ if B2 − 4AC = 0.(a) Circle(b) Ellipse(c) Hyperbola(d) Parabola Ax² + Bxy + Cy² + Dx+ Ey + F = 0
True or False The equation y2 = 9 + x2 is symmetric with respect to the x -axis, the y -axis, and the origin.
If (r, θ) are polar coordinates, the equation(a) Parabola(b) Hyperbola(c) Ellipse(d) Circle r = 2 2 + 3 sin 0 defines a(an)
Find the vertical asymptotes, if any, and the horizontal or oblique asymptote, if any, of y x x² - 9 4
If θ is acute, the Half-angle Formula for the cosine function is cosθ/2= _______.
In Problems 7–26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = 3t+2, y(t) = t + 1; 0≤ t ≤ 4
To graph y = (x − 3)2 + 1, shift the graph of y = x2 to the right _________ units and then _________ 1 unit.
f (x) = log4 (x − 2)(a) Solve f (x) = 2.(b) Solve f (x) ≤ 2.
In Problems 1 – 10, identify each equation. If it is a parabola, give its vertex, focus, and directrix; if it is an ellipse, give its center, vertices, and foci; if it is a hyperbola, give its
The graph of y = (x − 3)2 − 5 has vertex and axis of symmetry _________________.
For a hyperbola, the foci lie on a line called the_____________________.
For the parabola y2 = 4ax, the line segment joining the two points (a, 2a) and (a, −2a) is called the ___________ ____________ .
The equation ax2 + 6y2 + 12y = 0 defines an ellipse if ________________.(a) a < 0(b) a = 0(c) a > 0(d) a is any real number
Answer Problems 9 – 11 using the figure below.The equation of the hyperbola is of the form (h, k) YA | Transverse axis F₂ F₁ X
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = t - 3, y(t) = 2t+4; 0 < t < 2
In Problems 7–26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = t + 2, y(t) = √t; t 20
For an ellipse, the foci lie on a line called the ______________.(a) Minor axis(b) Major axis(c) Directrix(d) Latus rectum
Open the “Ellipse” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Interactive Figures) or at bit.ly/3raFUGB.(a) Check the box “Show
What is the domain of the function f(x) = 3 -? sin x + cos X
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = √2t, y(t) = 4t; t > 0
For the ellipsethe vertices are the points ______and _________. + 4 25 1,
In Problems 7 – 12, identify the conic defined by each polar equation. Also give the position of the directrix. r = 6 8 + 2 sin 0
Answer Problems 11–14 using the figure shownThe coordinates of the vertex are _______________-______________. VA F V=(3, 2) D X
For the ellipsethe value of a is _______, the value of b is _______, and the major axis is the ______-axis. x2 + 25 9 1,
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = √t + 4, y(t) = √t - 4; t ≥ 0
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = csct, y(t) = cott; ·sts. Is FIN
In Problems 21–30, find an equation for the hyperbola described. Graph the equation.Center at (0, 0); focus at (−3, 0); vertex at (2, 0)
Open the “Hyperbola” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Interactive Figures) or at bit.ly/3raFUGB. Check the boxes “Show
Answer Problems 11–14 using the figure shownIf a = 4, then the coordinates of the focus are ______. VA F V=(3, 2) D X
For the hyperbolathe asymptotes are________ and ________. y2 16 x2 81 1,
In Problems 15–18, the graph of an ellipse is given. Match each graph to its equation. (A) - + y² = 1 (B) x² + 2 || 1 (C) 16 = 1 (D)+ 16 = = 1
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x (t) = 3t², y(t)t + 1; -∞ < t < ∞
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = et, y(t) = et; t≥ 0
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = 2t4, y(t) = 4t²; ∞0 < t < ∞0
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = 2et, y(t) = 1 + et; t≥ 0
In Problems 13 – 24, analyze each equation and graph it. r = 12 4 + 8 sin 0
In Problems 15–22, the graph of a parabola is given. Match each graph to its equation. (A) y² = 4x (B) x² = 4y CA y2 = - 4x (D) x² = -4y (E) (y 1)² = 4(x - 1) (F) (x + 1)² = 4(y + 1) (G) (y -
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = √t, y(t) = t³/²; t > 0
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = t3/2+1, y(t) = √t; t > 0
In Problems 19–28, analyze each equation. That is, find the center, vertices, and foci of each ellipse and graph it. X y + 25 4 1
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x (t) 2 cost, y(t) = 3 sint; 0 ≤ t ≤ 2π
In Problems 15–22, the graph of a parabola is given. Match each graph to its equation. (A) y² = 4x (B) x² = 4y CA y2 = - 4x (D) x² = -4y (E) (y 1)² = 4(x - 1) (F) (x + 1)² = 4(y + 1) (G) (y -
In Problems 13 – 24, analyze each equation and graph it. r = 8 2 + 4 cos0
In Problems 13 – 24, analyze each equation and graph it. r = 8 2 - sin 0
In a hyperbola, if a = 3 and c = 5, then b = ____________.(a) 1(b) 2(c) 4(d) 8
In Problems 19–28, analyze each equation. That is, find the center, vertices, and foci of each ellipse and graph it. X 9 4 = 1
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = 2 cost, y(t) = 3 sint; 0 ≤ t ≤ π
In Problems 13 – 24, analyze each equation and graph it. r(32 sin0) = 6
In Problems 11 – 18, find an equation of the conic described. Graph the equation.Center at (−1, 2); a = 3; c = 4; transverse axis parallel to the x -axis
In Problems 19–28, analyze each equation. That is, find the center, vertices, and foci of each ellipse and graph it. X 9 25 1
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = 2 cost, y(t) = 3 sint; - < t ≤0
In Problems 15–22, the graph of a parabola is given. Match each graph to its equation. (A) y² = 4x (B) x² = 4y CA y2 = - 4x (D) x² = -4y (E) (y 1)² = 4(x - 1) (F) (x + 1)² = 4(y + 1) (G) (y -
In Problems 13 – 24, analyze each equation and graph it. r(2 cose) = 2 -
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. FIN x(t) = 2 cost, y(t) = sint; 0 ≤t≤ 12/12
In Problems 19–28, analyze each equation. That is, find the center, vertices, and foci of each ellipse and graph it. 4x² + y² 16
In Problems 19–28, analyze each equation. That is, find the center, vertices, and foci of each ellipse and graph it. +zx y2 16 1
In Problems 13 – 24, analyze each equation and graph it. r = 3 csc 0 csc 0 - 1
In Problems 13 – 24, analyze each equation and graph it. r = 6 sec0 2 sec 0 - 1
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t): = sect, y(t) = tant; 0≤t≤ 4
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = sin²t, y(t) = cos² t; 0 ≤ t ≤ 2π
In Problems 25 – 36, convert each polar equation to a rectangular equation. r = 3 1 - sin
In Problems 25 – 36, convert each polar equation to a rectangular equation. r = 8 4 + 3 sin
In Problems 19–28, analyze each equation. That is, find the center, vertices, and foci of each ellipse and graph it. 4y² + 9x² = 36 =
In Problems 7 – 26, graph the plane curve whose parametric equations are given, and show its orientation. Find the rectangular equation of each curve. x(t) = t², y(t) = Int; t > 0
In Problems 25 – 36, convert each polar equation to a rectangular equation. r = 10 5 + 4 cos0
In Problems 27 – 29, identify the conic that each polar equation represents, and graph it. r = 4 1 - cose
In Problems 27 – 29, identify the conic that each polar equation represents, and graph it. r = 2 6 sin
In Problems 25 – 36, convert each polar equation to a rectangular equation. r = 9 36 cos0
In Problems 21–30, find an equation for the hyperbola described. Graph the equation.Focus at (0, 6); vertices at (0, −2) and (0, 2)
In Problems 30 and 31 , convert each polar equation to a rectangular equation. r = 4 1 - cose
In Problems 25 – 36, convert each polar equation to a rectangular equation. r = 12 4 + 8 sin 0
In Problems 31–38, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. x2 25 - y2 9 = 1
In Problems 23–40, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation.Focus at (−2, 0); directrix the line x = 2
In Problems 25 – 36, convert each polar equation to a rectangular equation. r = 8 2 + 4 cos0
In Problems 30 and 31 , convert each polar equation to a rectangular equation. r = 8 4 + 8 cos0
In Problems 25 – 36, convert each polar equation to a rectangular equation. r = 8 2 sin 0
In Problems 21–30, find an equation for the hyperbola described. Graph the equation.Vertices at (−4, 0) and (4, 0); asymptote the line y = 2x
In Problems 23–40, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation.Focus at (0, −1); directrix the line y = 1
In Problems 21–30, find an equation for the hyperbola described. Graph the equation.Foci at (−4, 0) and (4, 0); asymptote the line y = −x
In Problems 31–38, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation. y² 16 x2 4 = 1
In Problems 29–40, find an equation for each ellipse. Graph the equation.Center at (0, 0); focus at (−1, 0); vertex at (3, 0)
In Problems 27 – 34, find two different pairs of parametric equations for each rectangular equation.x = y3/2
In Problems 25 – 36, convert each polar equation to a rectangular equation. r(32 sin0) = 6
Polar plots provide attainable speeds of a specific sailboat sailing at different angles to a wind of given speed. See the figure. Use the plot to approximate the attainable speed of the sailboat for
At 10:15 am, a radar station detects an aircraft at a point 80 miles away and 25 degrees north of due east. At 10:25 am, the aircraft is 110 miles away and 5 degrees south of due east.(a) Using the
The function f (x) = 3 sin(4x) has amplitude____________ and period ____________.
The Double-angle Formula for the sine function is sin(2θ) = ________________.
Show that r = a cos θ + b sin θ, with a, b not both zero, is the equation of a circle. Find the center and radius of the circle.
Express r2 = cos(2θ) in rectangular coordinates free of radicals.
Radar station A uses a coordinate system where A is located at the pole and due east is the polar axis. On this system, two other radar stations, B and C, are located at coordinates (150, −24°)
Prove that the area of the triangle with vertices (0, 0), (₁, ₁), and (r₂,0₂), 0≤ 0₁
Problems 95 – 104. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.
A 2-pound weight is attached to a 3-pound weight by a rope that passes over an ideal pulley. The smaller weight hangs vertically, while the larger weight sits on a frictionless inclined ramp with
Problems 94–103. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as
A 20-pound box sits at rest on a horizontal surface, and there is friction between the box and the surface. One side of the surface is raised slowly to create a ramp. The friction force f opposes the
Problems 94 – 103. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as
A box sitting on a horizontal surface is attached to a second box sitting on an inclined ramp by a rope that passes over an ideal pulley. The rope exerts a tension force T on both weights along the
Problems 95 – 104. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for later sections, a final exam, or subsequent courses such as calculus.

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