Question 1

Find the standard form of the equation of the parabola with the given characteristic and vertex at the origin. 
Vertical axis and passes through the point (-8,-8)

Accepted Answer

A)  $y ^ { 2 } = - 8 x$ 
B)  $y ^ { 2 } = x$ 
C)  $x ^ { 2 } = - 8 y$ 
D)  $x ^ { 2 } = 8 y$ 
E)  $y ^ { 2 } = 8 x$ 
A)  $y ^ { 2 } = - 8 x$ 
B)  $y ^ { 2 } = x$ 
C)  $x ^ { 2 } = - 8 y$ 
D)  $x ^ { 2 } = 8 y$ 
E)  $y ^ { 2 } = 8 x$

Question 2

A projectile is launched at a height of h feet above the ground at an angle of $\theta$ with the horizontal.The initial velocity is $v _ { 0 }$ feet per second, and the path of the projectile is modeled by the parametric equations $x = \left( v _ { 0 } \cos \theta \right) t \text { and } y = h + \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$ . Select the correct graph of the path of a projectile launched from ground level at the value of $\theta$ and $v _ { 0 }$ .
 $\theta = 45 ^ { \circ } , v _ { 0 } = 114$ feet per second

Accepted Answer

A)    
B)    
C)    
D)    
E)    
A)    
B)    
C)    
D)    
E)

Question 3

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.  $x = 3 ( t + 1 )$ $y = | t - 3 |$

Accepted Answer

A)  $y = \left| \frac { x } { 4 } - 3 \right|$ 
B)  $y = \left| \frac { x } { 3 } + 4 \right|$ 
C)  $y = | t - 4 |$ 
D)  $y = \left| \frac { x } { 3 } - 4 \right|$ 
E)  $y = | t + 4 |$ 
A)  $y = \left| \frac { x } { 4 } - 3 \right|$ 
B)  $y = \left| \frac { x } { 3 } + 4 \right|$ 
C)  $y = | t - 4 |$ 
D)  $y = \left| \frac { x } { 3 } - 4 \right|$ 
E)  $y = | t + 4 |$

Question 4

Find the inclination  (in degrees) of the line with a slope of m.Round your answer to one decimal places.  $m = - 3$

Accepted Answer

A)  $\Theta \approx 108.4 ^ { \circ }$ 
B)  $\Theta \approx 109.4 ^ { \circ }$ 
C)  $\Theta \approx 110.4 ^ { \circ }$ 
D)  $\Theta \approx 107.4 ^ { \circ }$ 
E)  $\Theta \approx 106.4 ^ { \circ }$ 
A)  $\Theta \approx 108.4 ^ { \circ }$ 
B)  $\Theta \approx 109.4 ^ { \circ }$ 
C)  $\Theta \approx 110.4 ^ { \circ }$ 
D)  $\Theta \approx 107.4 ^ { \circ }$ 
E)  $\Theta \approx 106.4 ^ { \circ }$

Question 5

Use the following results the polar equation of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is $r ^ { 2 } = \frac { b ^ { 2 } } { 1 - e ^ { 2 } \cos ^ { 2 } \theta }$ to write the polar form of the equation of the conic $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$ .

Accepted Answer

A)  $r ^ { 2 } = \frac { 25 } { 25 - 9 \cos ^ { 2 } \theta }$ 
B)  $r ^ { 2 } = \frac { 400 } { 25 - 9 \cos ^ { 2 } \theta }$ 
C)  $r ^ { 2 } = \frac { 400 } { 25 + 9 \cos ^ { 2 } \theta }$ 
D)  $r ^ { 2 } = \frac { 400 } { 9 + 25 \cos ^ { 2 } \theta }$ 
E)  $r ^ { 2 } = \frac { 400 } { 9 - 25 \cos ^ { 2 } \theta }$ 
A)  $r ^ { 2 } = \frac { 25 } { 25 - 9 \cos ^ { 2 } \theta }$ 
B)  $r ^ { 2 } = \frac { 400 } { 25 - 9 \cos ^ { 2 } \theta }$ 
C)  $r ^ { 2 } = \frac { 400 } { 25 + 9 \cos ^ { 2 } \theta }$ 
D)  $r ^ { 2 } = \frac { 400 } { 9 + 25 \cos ^ { 2 } \theta }$ 
E)  $r ^ { 2 } = \frac { 400 } { 9 - 25 \cos ^ { 2 } \theta }$

Question 6

Consider the equation $r = \sin k \theta$ .Select the correct graph of the equation for $k = 2.5$ .

Accepted Answer

A)    
B)   
C)   
D)    
E)    
A)    
B)   
C)   
D)    
E)

Question 7

Find the standard form of the equation of the ellipse with the given characteristics. 
Center: $( 0,4 )$ ; a = 2c; Vertices: $( - 8,8 ) , ( 8,8 )$ , where a - thr semimajor axis, b - semiminor axis and $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$ .

Accepted Answer

A)  $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$ 
B)  $\frac { x ^ { 2 } } { 64 } - \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } - \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$ 
D)  $\frac { x ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$ 
E)  $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 48 } = 1$ 
A)  $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$ 
B)  $\frac { x ^ { 2 } } { 64 } - \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } - \frac { ( y + 8 ) ^ { 2 } } { 48 } = 1$ 
D)  $\frac { x ^ { 2 } } { 64 } + \frac { ( y - 8 ) ^ { 2 } } { 48 } = 1$ 
E)  $\frac { ( x - 8 ) ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 48 } = 1$

Question 8

Find the standard form of the equation of the hyperbola with the given characteristics.
 vertices: (1, -2), (5, -2)
Asymptotes: $y = \frac { 5 } { 2 } x , y = - \frac { 5 } { 2 } x$

Accepted Answer

A)  $\frac { ( y - 3 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 3 ) ^ { 2 } } { 4 } + \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
C)  $\frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 3 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y + 3 ) ^ { 2 } } { 25 } = 1$ 
A)  $\frac { ( y - 3 ) ^ { 2 } } { 4 } - \frac { ( x + 2 ) ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 3 ) ^ { 2 } } { 4 } + \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
C)  $\frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 3 ) ^ { 2 } } { 4 } - \frac { ( y + 2 ) ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 2 ) ^ { 2 } } { 4 } - \frac { ( y + 3 ) ^ { 2 } } { 25 } = 1$

Question 9

Find the standard form of the equation of the hyperbola with the given characteristics. foci: $( \pm 4,0 )$ asymptotes: $y = \pm 2 x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 4 } = 1$ 
C)  $\frac { y ^ { 2 } } { \frac { 16 } { 5 } } - \frac { x ^ { 2 } } { \frac { 64 } { 5 } } = 1$ 
D)  $\frac { x ^ { 2 } } { \frac { 16 } { 5 } } - \frac { y ^ { 2 } } { \frac { 64 } { 5 } } = 1$ 
E)  $\frac { x ^ { 2 } } { \frac { 64 } { 5 } } - \frac { y ^ { 2 } } { \frac { 16 } { 5 } } = 1$ 
A)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 4 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 4 } = 1$ 
C)  $\frac { y ^ { 2 } } { \frac { 16 } { 5 } } - \frac { x ^ { 2 } } { \frac { 64 } { 5 } } = 1$ 
D)  $\frac { x ^ { 2 } } { \frac { 16 } { 5 } } - \frac { y ^ { 2 } } { \frac { 64 } { 5 } } = 1$ 
E)  $\frac { x ^ { 2 } } { \frac { 64 } { 5 } } - \frac { y ^ { 2 } } { \frac { 16 } { 5 } } = 1$

Question 10

Convert the polar equation to rectangular form.  $r ^ { 2 } = 7 \sin \theta$

Accepted Answer

A)  $x ^ { 2 } - y ^ { 2 } - \sqrt [ 3 ] { 49 } y ^ { 2 / 3 } = 0$ 
B)  $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 49 } y ^ { 2 / 3 } = 0$ 
C)  $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 7 } y ^ { 2 / 3 } = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 49 } y ^ { 2 / 3 } = 0$ 
E)  $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 7 } y ^ { 2 / 3 } = 0$ 
A)  $x ^ { 2 } - y ^ { 2 } - \sqrt [ 3 ] { 49 } y ^ { 2 / 3 } = 0$ 
B)  $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 49 } y ^ { 2 / 3 } = 0$ 
C)  $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 7 } y ^ { 2 / 3 } = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } - \sqrt [ 3 ] { 49 } y ^ { 2 / 3 } = 0$ 
E)  $x ^ { 2 } + y ^ { 2 } + \sqrt [ 3 ] { 7 } y ^ { 2 / 3 } = 0$

Question 11

Find the standard form of the equation of the ellipse with the given characteristics. 
Vertices: $( 0,4 ) , ( 8,4 )$ ; minor axis of length 10.

Accepted Answer

A)  $\frac { ( x + 4 ) ^ { 2 } } { 16 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { 16 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } - \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$ 
A)  $\frac { ( x + 4 ) ^ { 2 } } { 16 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { 16 } + \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } - \frac { ( y - 4 ) ^ { 2 } } { 25 } = 1$

Question 12

A projectile is launched at a height of h feet above the ground at an angle of $\theta$ with the horizontal.The initial velocity is $v _ { 0 }$ feet per second, and the path of the projectile is modeled by the parametric equations $x = \left( v _ { 0 } \cos \theta \right) t \text { and } y = h + \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$ . Select the correct graph of the path of a projectile launched from ground level at the value of $\theta$ and $v _ { 0 }$ .
$\theta = 60 ^ { \circ }$ , $v _ { 0 } = 93$ feet per second

Accepted Answer

A)    
B)    
C)    
D)   
E)    
A)    
B)    
C)    
D)   
E)

Question 13

Select the correct graph of the following equations of the hyperbola and find the center of the hyperbola.  $\frac { ( y + 3 ) ^ { 2 } } { 1 / 81 } - \frac { ( x - 2 ) ^ { 2 } } { 1 / 64 } = 1$

Accepted Answer

A) Center: (2, - 3)   
B) Center: (0, 0)   
C) Center: (2, - 3)   
D) Center: (0, 0)   
E) Center: (2, - 3)   
A) Center: (2, - 3)   
B) Center: (0, 0)   
C) Center: (2, - 3)   
D) Center: (0, 0)   
E) Center: (2, - 3)

Question 14

The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system.  $\theta = 90 ^ { \circ }$ , $( 2,2 )$

Accepted Answer

A)  $( - 2,0 )$ 
B)  $( 0,3 )$ 
C)  $( 2 , - 2 )$ 
D)  $( 2,0 )$ 
E)  $( - 2,2 )$ 
A)  $( - 2,0 )$ 
B)  $( 0,3 )$ 
C)  $( 2 , - 2 )$ 
D)  $( 2,0 )$ 
E)  $( - 2,2 )$

Question 15

Find the vertex and focus of the parabola. $x ^ { 2 } + 12 y = 0$

Accepted Answer

A) vertex: (0, 0) focus: $( 0 , - 3 )$ 
B) vertex: (0, 0) focus: $( 0,3 )$ 
C) vertex: $( 3,0 )$ focus: (0, 0) 
D) vertex: $( - 3,0 )$ focus: (0, 0) 
E) vertex: (0, 0) focus: $( 3,0 )$ 
A) vertex: (0, 0) focus: $( 0 , - 3 )$ 
B) vertex: (0, 0) focus: $( 0,3 )$ 
C) vertex: $( 3,0 )$ focus: (0, 0) 
D) vertex: $( - 3,0 )$ focus: (0, 0) 
E) vertex: (0, 0) focus: $( 3,0 )$

Question 16

Find the rectangular coordinates of the point given in polar coordinates.Round your results to two decimal places.  $( 5.8,2.62 )$

Accepted Answer

A) Rectangular coordinates: (-5.03,2.89) 
B) Rectangular coordinates: (5.80,2.89) 
C) Rectangular coordinates: (2.89,-5.03) 
D) Rectangular coordinates: (-4.03,3.89) 
E) Rectangular coordinates: (3.89,-4.03) 
A) Rectangular coordinates: (-5.03,2.89) 
B) Rectangular coordinates: (5.80,2.89) 
C) Rectangular coordinates: (2.89,-5.03) 
D) Rectangular coordinates: (-4.03,3.89) 
E) Rectangular coordinates: (3.89,-4.03)

Question 17

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. 
Vertices: (0,±7); focies: (0,±4)

Accepted Answer

A)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
B)  $\frac { x ^ { 2 } } { 33 } + \frac { y ^ { 2 } } { 49 } = 1$ 
C)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = - 1$ 
D)  $- \frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
E)  $\frac { x ^ { 2 } } { 33 } + \frac { y ^ { 2 } } { 49 } = - 1$ 
A)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
B)  $\frac { x ^ { 2 } } { 33 } + \frac { y ^ { 2 } } { 49 } = 1$ 
C)  $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = - 1$ 
D)  $- \frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ 
E)  $\frac { x ^ { 2 } } { 33 } + \frac { y ^ { 2 } } { 49 } = - 1$

Question 18

Find the center and vertices which located on the major axis of the ellipse from given equation and select its graph.  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 / 9 } = 1$

Accepted Answer

A) Center: (0,0)Vertices: (±3,0)   
B) Center: (0,0)Vertices: (-3,0)   
C) Center: (0,0)Vertices: (3,0)   
D) Center: (0,0)Vertices: (±3,0)   
E) Center: (0,0)Vertices: (-3,0)   
A) Center: (0,0)Vertices: (±3,0)   
B) Center: (0,0)Vertices: (-3,0)   
C) Center: (0,0)Vertices: (3,0)   
D) Center: (0,0)Vertices: (±3,0)   
E) Center: (0,0)Vertices: (-3,0)

Question 19

Select the correct graph of the polar equation.Find an interval for $\theta$ for which the graph is traced only once.  $r = 4 - 8 \cos \theta$

Accepted Answer

A)    $0 \leq \theta < 2 \pi$ 
B)    $0 \leq \theta < 2 \pi$ 
C)    $0 \leq \theta < 2 \pi$ 
D)    $0 \leq \theta < 2 \pi$ 
E)    $0 \leq \theta < 2 \pi$ 
A)    $0 \leq \theta < 2 \pi$ 
B)    $0 \leq \theta < 2 \pi$ 
C)    $0 \leq \theta < 2 \pi$ 
D)    $0 \leq \theta < 2 \pi$ 
E)    $0 \leq \theta < 2 \pi$

Question 20

Find the standard form of the equation of the parabola with the given characteristics. 
Vertex: $( 1,2 )$ ; directrix: $y = - 2$

Accepted Answer

A)  $( y - 1 ) ^ { 2 } = 2 ( x - 2 )$ 
B)  $( x - 1 ) ^ { 2 } = - 16 ( y - 2 )$ 
C)  $( y - 1 ) ^ { 2 } = - 16 ( x - 2 )$ 
D)  $( x - 1 ) ^ { 2 } = 2 ( y - 2 )$ 
E)  $( x - 1 ) ^ { 2 } = 16 ( y - 2 )$ 
A)  $( y - 1 ) ^ { 2 } = 2 ( x - 2 )$ 
B)  $( x - 1 ) ^ { 2 } = - 16 ( y - 2 )$ 
C)  $( y - 1 ) ^ { 2 } = - 16 ( x - 2 )$ 
D)  $( x - 1 ) ^ { 2 } = 2 ( y - 2 )$ 
E)  $( x - 1 ) ^ { 2 } = 16 ( y - 2 )$

Find the standard form of the equation of the parabola with the given characteristic and vertex at the origin. Vertical axis and passes through the point (-8,-8)

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = 3 ( t + 1 )$ $y = | t - 3 |$

Find the inclination (in degrees) of the line with a slope of m.Round your answer to one decimal places. $m = - 3$

Consider the equation $r = \sin k \theta$ .Select the correct graph of the equation for $k = 2.5$ .

Find the standard form of the equation of the ellipse with the given characteristics. Center: $( 0,4 )$ ; a = 2c; Vertices: $( - 8,8 ) , ( 8,8 )$ , where a - thr semimajor axis, b - semiminor axis and $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$ .

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (1, -2), (5, -2) Asymptotes: $y = \frac { 5 } { 2 } x , y = - \frac { 5 } { 2 } x$

Find the standard form of the equation of the hyperbola with the given characteristics. foci: $( \pm 4,0 )$ asymptotes: $y = \pm 2 x$

Convert the polar equation to rectangular form. $r ^ { 2 } = 7 \sin \theta$

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: $( 0,4 ) , ( 8,4 )$ ; minor axis of length 10.

Select the correct graph of the following equations of the hyperbola and find the center of the hyperbola. $\frac { ( y + 3 ) ^ { 2 } } { 1 / 81 } - \frac { ( x - 2 ) ^ { 2 } } { 1 / 64 } = 1$

The $x ^ { \prime } y ^ { \prime }$ -coordinate system has been rotated $\theta$ degrees from the $x y$ -coordinate system.The coordinates of a point in the $x y$ -coordinate system are given.Find the coordinates of the point in the rotated coordinate system. $\theta = 90 ^ { \circ }$ , $( 2,2 )$

Find the vertex and focus of the parabola. $x ^ { 2 } + 12 y = 0$

Find the rectangular coordinates of the point given in polar coordinates.Round your results to two decimal places. $( 5.8,2.62 )$

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,±7); focies: (0,±4)

Find the center and vertices which located on the major axis of the ellipse from given equation and select its graph. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 / 9 } = 1$

Select the correct graph of the polar equation.Find an interval for $\theta$ for which the graph is traced only once. $r = 4 - 8 \cos \theta$

Find the standard form of the equation of the parabola with the given characteristics. Vertex: $( 1,2 )$ ; directrix: $y = - 2$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

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Additional Topics In Trigonometery

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Limits and An Introduction To Calculus

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Exam 10: Topics In Analytic Geometry

Find the standard form of the equation of the parabola with the given characteristic and vertex at the origin. Vertical axis and passes through the point (-8,-8)

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. x=3(t+1)x = 3 ( t + 1 )x=3(t+1) y=∣t−3∣y = | t - 3 |y=∣t−3∣

Find the inclination (in degrees) of the line with a slope of m.Round your answer to one decimal places. m=−3m = - 3m=−3

Consider the equation r=sin⁡kθr = \sin k \thetar=sinkθ .Select the correct graph of the equation for k=2.5k = 2.5k=2.5 .

Find the standard form of the equation of the ellipse with the given characteristics. Center: (0,4)( 0,4 )(0,4) ; a = 2c; Vertices: (−8,8),(8,8)( - 8,8 ) , ( 8,8 )(−8,8),(8,8) , where a - thr semimajor axis, b - semiminor axis and c2=a2−b2c ^ { 2 } = a ^ { 2 } - b ^ { 2 }c2=a2−b2 .

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (1, -2), (5, -2) Asymptotes: y=52x,y=−52xy = \frac { 5 } { 2 } x , y = - \frac { 5 } { 2 } xy=25​x,y=−25​x

Find the standard form of the equation of the hyperbola with the given characteristics. foci: (±4,0)( \pm 4,0 )(±4,0) asymptotes: y=±2xy = \pm 2 xy=±2x

Convert the polar equation to rectangular form. r2=7sin⁡θr ^ { 2 } = 7 \sin \thetar2=7sinθ

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: (0,4),(8,4)( 0,4 ) , ( 8,4 )(0,4),(8,4) ; minor axis of length 10.

Select the correct graph of the following equations of the hyperbola and find the center of the hyperbola. (y+3)21/81−(x−2)21/64=1\frac { ( y + 3 ) ^ { 2 } } { 1 / 81 } - \frac { ( x - 2 ) ^ { 2 } } { 1 / 64 } = 11/81(y+3)2​−1/64(x−2)2​=1

Find the vertex and focus of the parabola. x2+12y=0x ^ { 2 } + 12 y = 0x2+12y=0

Find the rectangular coordinates of the point given in polar coordinates.Round your results to two decimal places. (5.8,2.62)( 5.8,2.62 )(5.8,2.62)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,±7); focies: (0,±4)

Find the center and vertices which located on the major axis of the ellipse from given equation and select its graph. x29+y21/9=1\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 / 9 } = 19x2​+1/9y2​=1

Select the correct graph of the polar equation.Find an interval for θ\thetaθ for which the graph is traced only once. r=4−8cos⁡θr = 4 - 8 \cos \thetar=4−8cosθ

Find the standard form of the equation of the parabola with the given characteristics. Vertex: (1,2)( 1,2 )(1,2) ; directrix: y=−2y = - 2y=−2

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Analytic Geometry In Three Dimensions

Limits and An Introduction To Calculus

Filters

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = 3 ( t + 1 )$ $y = | t - 3 |$

Find the inclination (in degrees) of the line with a slope of m.Round your answer to one decimal places. $m = - 3$

Consider the equation $r = \sin k \theta$ .Select the correct graph of the equation for $k = 2.5$ .

Find the standard form of the equation of the ellipse with the given characteristics. Center: $( 0,4 )$ ; a = 2c; Vertices: $( - 8,8 ) , ( 8,8 )$ , where a - thr semimajor axis, b - semiminor axis and $c ^ { 2 } = a ^ { 2 } - b ^ { 2 }$ .

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (1, -2), (5, -2) Asymptotes: $y = \frac { 5 } { 2 } x , y = - \frac { 5 } { 2 } x$

Find the standard form of the equation of the hyperbola with the given characteristics. foci: $( \pm 4,0 )$ asymptotes: $y = \pm 2 x$

Convert the polar equation to rectangular form. $r ^ { 2 } = 7 \sin \theta$

Find the standard form of the equation of the ellipse with the given characteristics. Vertices: $( 0,4 ) , ( 8,4 )$ ; minor axis of length 10.

Select the correct graph of the following equations of the hyperbola and find the center of the hyperbola. $\frac { ( y + 3 ) ^ { 2 } } { 1 / 81 } - \frac { ( x - 2 ) ^ { 2 } } { 1 / 64 } = 1$

Find the vertex and focus of the parabola. $x ^ { 2 } + 12 y = 0$

Find the rectangular coordinates of the point given in polar coordinates.Round your results to two decimal places. $( 5.8,2.62 )$

Find the center and vertices which located on the major axis of the ellipse from given equation and select its graph. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 1 / 9 } = 1$

Select the correct graph of the polar equation.Find an interval for $\theta$ for which the graph is traced only once. $r = 4 - 8 \cos \theta$

Find the standard form of the equation of the parabola with the given characteristics. Vertex: $( 1,2 )$ ; directrix: $y = - 2$