Question 1

Use a graphing utility to graph the conic.Determine the angle $\theta$ through which the axes are rotated.  $x ^ { 2 } + 2 x y + y ^ { 2 } = 25$

Accepted Answer

A)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
B)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
C)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
D)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
E)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
A)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
B)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
C)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
D)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$   
E)  $\theta = \frac { \pi } { 4 } \text { or } 45 ^ { \circ }$ , $y = - x \pm \sqrt { 25 }$

Question 2

Find the standard form of the equation of the ellipse with the given characteristics. 
Foci: $( - 5 , - 1 ) , ( - 5,5 )$ , endpoints of the major axis: $( - 5 , - 5 ) , ( - 5,9 )$

Accepted Answer

A)  $\frac { ( x - 5 ) ^ { 2 } } { 40 } + \frac { ( y + 2 ) ^ { 2 } } { 49 } = 1$ 
B)  $\frac { ( x + 5 ) ^ { 2 } } { 40 } + \frac { ( y - 2 ) ^ { 2 } } { 49 } = 1$ 
C)  $\frac { ( x + 5 ) ^ { 2 } } { 49 } + \frac { ( y - 2 ) ^ { 2 } } { 40 } = 1$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 49 } + \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
E)  $\frac { ( x + 2 ) ^ { 2 } } { 49 } + \frac { ( y - 5 ) ^ { 2 } } { 40 } = 1$ 
A)  $\frac { ( x - 5 ) ^ { 2 } } { 40 } + \frac { ( y + 2 ) ^ { 2 } } { 49 } = 1$ 
B)  $\frac { ( x + 5 ) ^ { 2 } } { 40 } + \frac { ( y - 2 ) ^ { 2 } } { 49 } = 1$ 
C)  $\frac { ( x + 5 ) ^ { 2 } } { 49 } + \frac { ( y - 2 ) ^ { 2 } } { 40 } = 1$ 
D)  $\frac { ( x - 2 ) ^ { 2 } } { 49 } + \frac { ( y + 5 ) ^ { 2 } } { 40 } = 1$ 
E)  $\frac { ( x + 2 ) ^ { 2 } } { 49 } + \frac { ( y - 5 ) ^ { 2 } } { 40 } = 1$

Question 3

Select the graph of the following equation, showing both sets of axes.  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 8 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 8 } = 1$

Accepted Answer

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

Question 4

Select the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.  $y ^ { 2 } - 9 y - 4 x - 21 = 0$

Accepted Answer

A) Parabola 
B) Circle 
C) Hyperbola 
D) Ellipse 
E) None of the above 
A) Parabola 
B) Circle 
C) Hyperbola 
D) Ellipse 
E) None of the above

Question 5

Give the standard form of the equation of the parabola with the given characteristics. vertex: (-7, -9) directrix: $x = - 9$

Accepted Answer

A)  $( x - 7 ) ^ { 2 } = - 8 ( y - 9 )$ 
B)  $( y - 9 ) ^ { 2 } = - 8 ( x - 7 )$ 
C)  $( y + 9 ) ^ { 2 } = 8 ( x + 7 )$ 
D)  $( y - 9 ) ^ { 2 } = 8 ( x - 7 )$ 
E)  $( x + 7 ) ^ { 2 } = - 8 ( y + 9 )$ 
A)  $( x - 7 ) ^ { 2 } = - 8 ( y - 9 )$ 
B)  $( y - 9 ) ^ { 2 } = - 8 ( x - 7 )$ 
C)  $( y + 9 ) ^ { 2 } = 8 ( x + 7 )$ 
D)  $( y - 9 ) ^ { 2 } = 8 ( x - 7 )$ 
E)  $( x + 7 ) ^ { 2 } = - 8 ( y + 9 )$

Question 6

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.  $x = 1 + 3 \cos \theta$ $y = 1 + 5 \sin \theta$

Accepted Answer

A)  $\frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 1 ) ^ { 2 } } { 9 } - \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$ 
A)  $\frac { ( x - 1 ) ^ { 2 } } { 9 } + \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$ 
B)  $\frac { ( x - 1 ) ^ { 2 } } { 9 } - \frac { ( y - 1 ) ^ { 2 } } { 25 } = 1$ 
C)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 25 } = 1$ 
D)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ 
E)  $\frac { ( x - 1 ) ^ { 2 } } { 25 } - \frac { ( y - 1 ) ^ { 2 } } { 9 } = 1$

Question 7

Write the equation of the ellipse that has its center at the origin, focus at (1, 0) and vertex at (9, 0).

Accepted Answer

A)  $\frac { x ^ { 2 } } { 80 } + \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 80 } = - 1$ 
D)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 80 } = 1$ 
E)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 80 } = 1$ 
A)  $\frac { x ^ { 2 } } { 80 } + \frac { y ^ { 2 } } { 81 } = 1$ 
B)  $\frac { x ^ { 2 } } { 1 } + \frac { y ^ { 2 } } { 9 } = 1$ 
C)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 80 } = - 1$ 
D)  $\frac { x ^ { 2 } } { 81 } - \frac { y ^ { 2 } } { 80 } = 1$ 
E)  $\frac { x ^ { 2 } } { 81 } + \frac { y ^ { 2 } } { 80 } = 1$

Question 8

Describe the graph of the polar equation and find the corresponding rectangular equation.Select the correct graph.  $r = 2$

Accepted Answer

A) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = - 4$   
B) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 4$   
C) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 2$   
D) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 4$   
E) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 2$   
A) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = - 4$   
B) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 4$   
C) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 2$   
D) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 4$   
E) The graph of the polar equation consists of all points that are two units from the pole. $x ^ { 2 } + y ^ { 2 } = 2$

Question 9

Select the graph of the equation.  $r = 5 \sec \left( \theta - \frac { \pi } { 2 } \right)$

Accepted Answer

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

Question 10

Find a polar equation of the conic with its focus at the pole. 
Conics$\quad$$\quad$Vertex or vertices
Parabola$\quad$$\quad$( 4,0 )

Accepted Answer

A)  $\frac { 8 } { 1 - \cos \theta }$ 
B)  $\frac { 8 } { 1 - \sin \theta }$ 
C)  $\frac { - 8 } { 1 - \sin \theta }$ 
D)  $\frac { 8 } { 1 + \sin \theta }$ 
E)  $\frac { 8 } { 1 + \cos \theta }$ 
A)  $\frac { 8 } { 1 - \cos \theta }$ 
B)  $\frac { 8 } { 1 - \sin \theta }$ 
C)  $\frac { - 8 } { 1 - \sin \theta }$ 
D)  $\frac { 8 } { 1 + \sin \theta }$ 
E)  $\frac { 8 } { 1 + \cos \theta }$

Question 11

Roads are often designed with parabolic surfaces to allow rain to drain off.A particular road that is 44 feet wide is 0.4 foot higher in the center than it is on the sides (see figure).    
Where $a = 44$ ft, $b = 0.4 \mathrm { ft }$
Find an equation of the parabola that models the road surface.(Assume that the origin is at the center of the road.)

Accepted Answer

A)  $x ^ { 2 } = - 1210 y \text { or } y = - \frac { 1 } { 1210 } x ^ { 2 }$ 
B)  $y ^ { 2 } = - 1210 x \text { or } x = - \frac { 1 } { 1210 } y ^ { 2 }$ 
C)  $x ^ { 2 } = 1210 y \text { or } y = \frac { 1 } { 1210 } x ^ { 2 }$ 
D)  $y ^ { 2 } = 1210 \mathrm { x } \text { or } x = \frac { 1 } { 1210 } y ^ { 2 }$ 
E)  $x ^ { 2 } = 44 y \text { or } y = \frac { 1 } { 44 } x ^ { 2 }$ 
A)  $x ^ { 2 } = - 1210 y \text { or } y = - \frac { 1 } { 1210 } x ^ { 2 }$ 
B)  $y ^ { 2 } = - 1210 x \text { or } x = - \frac { 1 } { 1210 } y ^ { 2 }$ 
C)  $x ^ { 2 } = 1210 y \text { or } y = \frac { 1 } { 1210 } x ^ { 2 }$ 
D)  $y ^ { 2 } = 1210 \mathrm { x } \text { or } x = \frac { 1 } { 1210 } y ^ { 2 }$ 
E)  $x ^ { 2 } = 44 y \text { or } y = \frac { 1 } { 44 } x ^ { 2 }$

Question 12

Find the standard form of the equation of the hyperbola with the given characteristics. focies: (±4,0), asymptotes: $y = \pm 5 x$

Accepted Answer

A)  $\frac { x ^ { 2 } } { \frac { 1 } { 26 } } - \frac { y ^ { 2 } } { \frac { 1 } { 26 } } = 1$ 
B)  $\frac { x ^ { 2 } } { \frac { 8 } { 13 } } - \frac { y ^ { 2 } } { \frac { 200 } { 13 } } = 1$ 
C)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 400 } = 1$ 
E)  $\frac { x ^ { 2 } } { \frac { 200 } { 13 } } - \frac { y ^ { 2 } } { \frac { 8 } { 13 } } = 1$ 
A)  $\frac { x ^ { 2 } } { \frac { 1 } { 26 } } - \frac { y ^ { 2 } } { \frac { 1 } { 26 } } = 1$ 
B)  $\frac { x ^ { 2 } } { \frac { 8 } { 13 } } - \frac { y ^ { 2 } } { \frac { 200 } { 13 } } = 1$ 
C)  $\frac { y ^ { 2 } } { 25 } - \frac { x ^ { 2 } } { 16 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 400 } = 1$ 
E)  $\frac { x ^ { 2 } } { \frac { 200 } { 13 } } - \frac { y ^ { 2 } } { \frac { 8 } { 13 } } = 1$

Question 13

Convert the polar equation to rectangular form.  $r = 2 \sin \theta$

Accepted Answer

A)  $x ^ { 2 } + y ^ { 2 } - 2 y = 0$ 
B)  $x ^ { 2 } + y ^ { 2 } + 2 y = 0$ 
C)  $x ^ { 2 } + y ^ { 2 } - 2 \mathrm { x } = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } + 2 \mathrm { x } = 0$ 
E)  $x ^ { 2 } + y ^ { 2 } - 2 = 0$ 
A)  $x ^ { 2 } + y ^ { 2 } - 2 y = 0$ 
B)  $x ^ { 2 } + y ^ { 2 } + 2 y = 0$ 
C)  $x ^ { 2 } + y ^ { 2 } - 2 \mathrm { x } = 0$ 
D)  $x ^ { 2 } + y ^ { 2 } + 2 \mathrm { x } = 0$ 
E)  $x ^ { 2 } + y ^ { 2 } - 2 = 0$

Question 14

Select the curve represented by the parametric equations.  $\begin{array} { l } 
x = 4 ( t + 1 ) \
y = | t - 4 |
\end{array}$

Accepted Answer

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

Question 15

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form.  $x y + 3 = 0$

Accepted Answer

A)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 3 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 3 } = 1$ 
B)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 3 } = 1$ 
C)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$ 
D)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 3 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$ 
E)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$ 
A)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 3 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 3 } = 1$ 
B)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 3 } = 1$ 
C)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$ 
D)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 3 } - \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$ 
E)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 6 } = 1$

Question 16

Use the discriminant to classify the graph.  $x ^ { 2 } - 10 x y - 8 y ^ { 2 } - 18 = 0$

Accepted Answer

A) The graph is a parabola. 
B) The graph is a hyperbola. 
C) The graph is a ellipse. 
D) The graph is a cone. 
E) The graph is a circle. 
A) The graph is a parabola. 
B) The graph is a hyperbola. 
C) The graph is a ellipse. 
D) The graph is a cone. 
E) The graph is a circle.

Question 17

Find the standard form of the equation of the ellipse with vertices $( 0 , \pm 8 )$ and eccentricity $e = \frac { 3 } { 8 }$ .

Accepted Answer

A)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 55 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$ 
E)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$ 
A)  $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 64 } = 1$ 
B)  $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 55 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 9 } = 1$ 
E)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 55 } = 1$

Question 18

Find the vertex and focus of the parabola from the given equation and select its graph.  $2 x + y ^ { 2 } = 0$

Accepted Answer

A) Vertex: (0,0)Focus: (- $\frac { 1 } { 2 }$ ,0)   
B) Vertex: (- $\frac { 1 } { 2 }$ ,0)Focus: (0,0)   
C) Vertex: (0,0)Focus: (0, $\frac { 1 } { 2 }$ )   
D) Vertex: (0,0)Focus: (0,- $\frac { 1 } { 2 }$ )   
E) Vertex: (0,0)Focus: ( $\frac { 1 } { 2 }$ ,0)   
A) Vertex: (0,0)Focus: (- $\frac { 1 } { 2 }$ ,0)   
B) Vertex: (- $\frac { 1 } { 2 }$ ,0)Focus: (0,0)   
C) Vertex: (0,0)Focus: (0, $\frac { 1 } { 2 }$ )   
D) Vertex: (0,0)Focus: (0,- $\frac { 1 } { 2 }$ )   
E) Vertex: (0,0)Focus: ( $\frac { 1 } { 2 }$ ,0)

Question 19

Select the curve represented by the parametric equations. 
Cycloid: $x = \Theta + \sin \Theta , y = 4 - \cos \Theta $

Accepted Answer

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

Question 20

Select the curve represented by the parametric equations.(indicate the orientation of the curve) $\begin{array} { l } 
x = 3 - 4 t \
y = 4 + 3 t
\end{array}$

Accepted Answer

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

Use a graphing utility to graph the conic.Determine the angle $\theta$ through which the axes are rotated. $x ^ { 2 } + 2 x y + y ^ { 2 } = 25$

Find the standard form of the equation of the ellipse with the given characteristics. Foci: $( - 5 , - 1 ) , ( - 5,5 )$ , endpoints of the major axis: $( - 5 , - 5 ) , ( - 5,9 )$

Select the graph of the following equation, showing both sets of axes. $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 8 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 8 } = 1$

Select the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } - 9 y - 4 x - 21 = 0$

Give the standard form of the equation of the parabola with the given characteristics. vertex: (-7, -9) directrix: $x = - 9$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = 1 + 3 \cos \theta$ $y = 1 + 5 \sin \theta$

Write the equation of the ellipse that has its center at the origin, focus at (1, 0) and vertex at (9, 0).

Describe the graph of the polar equation and find the corresponding rectangular equation.Select the correct graph. $r = 2$

Select the graph of the equation. $r = 5 \sec \left( \theta - \frac { \pi } { 2 } \right)$

Find a polar equation of the conic with its focus at the pole. Conics $\quad$ $\quad$ Vertex or vertices Parabola $\quad$ $\quad$ ( 4,0 )

Find the standard form of the equation of the hyperbola with the given characteristics. focies: (±4,0), asymptotes: $y = \pm 5 x$

Convert the polar equation to rectangular form. $r = 2 \sin \theta$

Select the curve represented by the parametric equations. x=4(t+1) y=|t-4|

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form. $x y + 3 = 0$

Use the discriminant to classify the graph. $x ^ { 2 } - 10 x y - 8 y ^ { 2 } - 18 = 0$

Find the standard form of the equation of the ellipse with vertices $( 0 , \pm 8 )$ and eccentricity $e = \frac { 3 } { 8 }$ .

Find the vertex and focus of the parabola from the given equation and select its graph. $2 x + y ^ { 2 } = 0$

Select the curve represented by the parametric equations. Cycloid: $x = \Theta + \sin \Theta , y = 4 - \cos \Theta$

Select the curve represented by the parametric equations.(indicate the orientation of the curve) x=3-4t y=4+3t

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

Exam 10: Topics In Analytic Geometry

Use a graphing utility to graph the conic.Determine the angle θ\thetaθ through which the axes are rotated. x2+2xy+y2=25x ^ { 2 } + 2 x y + y ^ { 2 } = 25x2+2xy+y2=25

Find the standard form of the equation of the ellipse with the given characteristics. Foci: (−5,−1),(−5,5)( - 5 , - 1 ) , ( - 5,5 )(−5,−1),(−5,5) , endpoints of the major axis: (−5,−5),(−5,9)( - 5 , - 5 ) , ( - 5,9 )(−5,−5),(−5,9)

Select the graph of the following equation, showing both sets of axes. (x′)28−(y′)28=1\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 8 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 8 } = 18(x′)2​−8(y′)2​=1

Select the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. y2−9y−4x−21=0y ^ { 2 } - 9 y - 4 x - 21 = 0y2−9y−4x−21=0

Give the standard form of the equation of the parabola with the given characteristics. vertex: (-7, -9) directrix: x=−9x = - 9x=−9

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. x=1+3cos⁡θx = 1 + 3 \cos \thetax=1+3cosθ y=1+5sin⁡θy = 1 + 5 \sin \thetay=1+5sinθ

Write the equation of the ellipse that has its center at the origin, focus at (1, 0) and vertex at (9, 0).

Describe the graph of the polar equation and find the corresponding rectangular equation.Select the correct graph. r=2r = 2r=2

Select the graph of the equation. r=5sec⁡(θ−π2)r = 5 \sec \left( \theta - \frac { \pi } { 2 } \right)r=5sec(θ−2π​)

Find a polar equation of the conic with its focus at the pole. Conics \quad\quad Vertex or vertices Parabola \quad\quad ( 4,0 )

Find the standard form of the equation of the hyperbola with the given characteristics. focies: (±4,0), asymptotes: y=±5xy = \pm 5 xy=±5x

Convert the polar equation to rectangular form. r=2sin⁡θr = 2 \sin \thetar=2sinθ

Select the curve represented by the parametric equations. x=4(t+1) y=|t-4|

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form. xy+3=0x y + 3 = 0xy+3=0

Use the discriminant to classify the graph. x2−10xy−8y2−18=0x ^ { 2 } - 10 x y - 8 y ^ { 2 } - 18 = 0x2−10xy−8y2−18=0

Find the standard form of the equation of the ellipse with vertices (0,±8)( 0 , \pm 8 )(0,±8) and eccentricity e=38e = \frac { 3 } { 8 }e=83​ .

Find the vertex and focus of the parabola from the given equation and select its graph. 2x+y2=02 x + y ^ { 2 } = 02x+y2=0

Select the curve represented by the parametric equations. Cycloid: x=Θ+sin⁡Θ,y=4−cos⁡Θx = \Theta + \sin \Theta , y = 4 - \cos \Theta x=Θ+sinΘ,y=4−cosΘ

Select the curve represented by the parametric equations.(indicate the orientation of the curve) x=3-4t y=4+3t

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

Use a graphing utility to graph the conic.Determine the angle $\theta$ through which the axes are rotated. $x ^ { 2 } + 2 x y + y ^ { 2 } = 25$

Find the standard form of the equation of the ellipse with the given characteristics. Foci: $( - 5 , - 1 ) , ( - 5,5 )$ , endpoints of the major axis: $( - 5 , - 5 ) , ( - 5,9 )$

Select the graph of the following equation, showing both sets of axes. $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 8 } - \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 8 } = 1$

Select the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } - 9 y - 4 x - 21 = 0$

Give the standard form of the equation of the parabola with the given characteristics. vertex: (-7, -9) directrix: $x = - 9$

Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. $x = 1 + 3 \cos \theta$ $y = 1 + 5 \sin \theta$

Describe the graph of the polar equation and find the corresponding rectangular equation.Select the correct graph. $r = 2$

Select the graph of the equation. $r = 5 \sec \left( \theta - \frac { \pi } { 2 } \right)$

Find a polar equation of the conic with its focus at the pole. Conics $\quad$ $\quad$ Vertex or vertices Parabola $\quad$ $\quad$ ( 4,0 )

Find the standard form of the equation of the hyperbola with the given characteristics. focies: (±4,0), asymptotes: $y = \pm 5 x$

Convert the polar equation to rectangular form. $r = 2 \sin \theta$

Rotate the axes to eliminate the xy-term in the equation.Then write the equation in standard form. $x y + 3 = 0$

Use the discriminant to classify the graph. $x ^ { 2 } - 10 x y - 8 y ^ { 2 } - 18 = 0$

Find the standard form of the equation of the ellipse with vertices $( 0 , \pm 8 )$ and eccentricity $e = \frac { 3 } { 8 }$ .

Find the vertex and focus of the parabola from the given equation and select its graph. $2 x + y ^ { 2 } = 0$

Select the curve represented by the parametric equations. Cycloid: $x = \Theta + \sin \Theta , y = 4 - \cos \Theta$