Question 1

Use the Quadratic Formula to solve for y in the following equation. $36 x ^ { 2 } - 84 x y + 49 y ^ { 2 } - 70 x - 30 y = 0$

Accepted Answer

A)  $\frac { - 42 x - 15 \pm \sqrt { - 84 x + 3655 } } { 49 }$ 
B)  $\frac { 2 ( 42 x + 15 ) \pm \sqrt { - 84 x + 14620 } } { 49 }$ 
C)  $\frac { 42 x + 15 \pm \sqrt { 4690 x + 225 } } { 49 }$ 
D)  $\frac { 84 x + 30 \pm \sqrt { - 42 x + 7085 } } { 98 }$ 
E)  $\frac { - 84 x - 30 \pm \sqrt { - 42 x + 4330 } } { 98 }$ 
A)  $\frac { - 42 x - 15 \pm \sqrt { - 84 x + 3655 } } { 49 }$ 
B)  $\frac { 2 ( 42 x + 15 ) \pm \sqrt { - 84 x + 14620 } } { 49 }$ 
C)  $\frac { 42 x + 15 \pm \sqrt { 4690 x + 225 } } { 49 }$ 
D)  $\frac { 84 x + 30 \pm \sqrt { - 42 x + 7085 } } { 98 }$ 
E)  $\frac { - 84 x - 30 \pm \sqrt { - 42 x + 4330 } } { 98 }$

Question 2

Find the standard form of the parabola with the given characteristic and vertex at the origin.
focus: $( 0 , - 5 )$

Accepted Answer

A)  $x ^ { 2 } = - 20 y$ 
B)  $x ^ { 2 } = - 5 y$ 
C)  $x ^ { 2 } = 5 y$ 
D)  $y ^ { 2 } = - 20 x$ 
E)  $y ^ { 2 } = - 5 x$ 
A)  $x ^ { 2 } = - 20 y$ 
B)  $x ^ { 2 } = - 5 y$ 
C)  $x ^ { 2 } = 5 y$ 
D)  $y ^ { 2 } = - 20 x$ 
E)  $y ^ { 2 } = - 5 x$

Question 3

The graph of the following polar equation is a circle. Sketch its graph, and find its radius.
$$r = 5 \sin ( \theta )$$

Accepted Answer

A) 
 
radius: 5 
B) 
 
radius: $2.5$ 
C) 
 
radius: 5 
D) 
 
radius: 5 
E) 
 
radius: 25 
A) 
 
radius: 5 
B) 
 
radius: $2.5$ 
C) 
 
radius: 5 
D) 
 
radius: 5 
E) 
 
radius: 25

Question 4

Solve the following system of quadratic equations algebraically by the method of substitution.
$$\left\{ \begin{array} { l } 
8 x ^ { 2 } - 6 y ^ { 2 } + 80 = 0 \
8 x + 4 y = 0
\end{array} \right.$$

Accepted Answer

A)  $( \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 7 } , - 2 \sqrt { 7 } )$ 
B)  $( - \sqrt { 7 } , 2 \sqrt { 7 } ) , ( \sqrt { 7 } , - 2 \sqrt { 7 } )$ 
C)  $( - \sqrt { 7 } , - 2 \sqrt { 7 } ) , ( \sqrt { 7 } , - 2 \sqrt { 7 } )$ 
D)  $( - \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 5 } , 2 \sqrt { 5 } )$ 
E)  $( - \sqrt { 5 } , 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
A)  $( \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 7 } , - 2 \sqrt { 7 } )$ 
B)  $( - \sqrt { 7 } , 2 \sqrt { 7 } ) , ( \sqrt { 7 } , - 2 \sqrt { 7 } )$ 
C)  $( - \sqrt { 7 } , - 2 \sqrt { 7 } ) , ( \sqrt { 7 } , - 2 \sqrt { 7 } )$ 
D)  $( - \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 5 } , 2 \sqrt { 5 } )$ 
E)  $( - \sqrt { 5 } , 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$

Question 5

Find the standard form of the equation of the hyperbola with the given characteristics.
vertices: $( 0 , \pm 4 ) \quad$ foci: $( 0 , \pm 8 )$

Accepted Answer

A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 64 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 64$ 
E)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 64$ 
A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 64 } = 1$ 
B)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 1$ 
C)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 64$ 
E)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 48 } = 64$

Question 6

Find a polar equation of the conic with the given characteristics and with one focus at the pole.
$\begin{array} { l c c } \text { Conic } & \text { Eccentricity } & \text { Directrix } \ \text { Ellipse } & e = \frac { 1 } { 2 } & y = 5 \end{array}$

Accepted Answer

A)  $r = \frac { 1 } { 2 + \cos \theta }$ 
B)  $r = \frac { 5 } { 2 + \cos \theta }$ 
C)  $r = \frac { 5 } { 2 - \cos \theta }$ 
D)  $r = \frac { 5 } { 2 - \sin \theta }$ 
E)  $r = \frac { 5 } { 2 + \sin \theta }$ 
A)  $r = \frac { 1 } { 2 + \cos \theta }$ 
B)  $r = \frac { 5 } { 2 + \cos \theta }$ 
C)  $r = \frac { 5 } { 2 - \cos \theta }$ 
D)  $r = \frac { 5 } { 2 - \sin \theta }$ 
E)  $r = \frac { 5 } { 2 + \sin \theta }$

Question 7

Find the vertices and asymptotes of the hyperbola.
$$9 y ^ { 2 } - 4 x ^ { 2 } = 36$$

Accepted Answer

A)  vertices: $( 0 , \pm 2 ) \quad$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
B)  vertices: $( 0 , \pm 2 ) \quad$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
C)  vertices: $( \pm 2,0 ) \quad$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
D)  vertices: $( \pm 2,0 ) \quad$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
E)  vertices: $( \pm 2,3 ) \quad$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
A)  vertices: $( 0 , \pm 2 ) \quad$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
B)  vertices: $( 0 , \pm 2 ) \quad$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
C)  vertices: $( \pm 2,0 ) \quad$ asymptote: $y = \pm \frac { 2 } { 3 } x$ 
D)  vertices: $( \pm 2,0 ) \quad$ asymptote: $y = \pm \frac { 3 } { 2 } x$ 
E)  vertices: $( \pm 2,3 ) \quad$ asymptote: $y = \pm \frac { 2 } { 3 } x$

Question 8

Sketch the graph of the polar equation using symmetry, zeros, maximum $r$-values, and any other additional points.
$$r ^ { 2 } = 4 \sin 2 \theta$$
Use either grid below for your graph, whichever is more convenient.

Accepted Answer

symmetry: with respe

Question 9

Test for symmetry with respect to $\theta = \pi / 2$, the polar axis, and the pole.
$$r = \frac { 6 } { 5 + \sin \theta }$$

Accepted Answer

A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$ 
A)  symmetric with respect to the pole 
B)  symmetric with respect to the polar axis 
C)  symmetric with respect to $\theta = \pi / 2$ 
D)  symmetric with respect to the pole and the polar axis 
E)  symmetric with respect to the pole and $\theta = \pi / 2$

Question 10

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

Accepted Answer

A)  $r = \frac { 2 } { 1 - 2 \sin \theta }$ 
B)  $r = \frac { 2 } { 1 + 2 \sin \theta }$ 
C)  $r = \frac { 2 } { 1 + 2 \cos \theta }$ 
D)  $r = \frac { 2 } { 1 - 2 \cos \theta }$ 
E)  $r = \frac { 2 } { 1 + 1 \cos \theta }$ 
A)  $r = \frac { 2 } { 1 - 2 \sin \theta }$ 
B)  $r = \frac { 2 } { 1 + 2 \sin \theta }$ 
C)  $r = \frac { 2 } { 1 + 2 \cos \theta }$ 
D)  $r = \frac { 2 } { 1 - 2 \cos \theta }$ 
E)  $r = \frac { 2 } { 1 + 1 \cos \theta }$

Question 11

Find the vertex and focus of the parabola below.
$$y ^ { 2 } + 6 y - 8 x + 20 = 0$$

Accepted Answer

A) 
Vertex: $\left( - \frac { 11 } { 8 } , 3 \right)$
Focus: $\left( \frac { 5 } { 8 } , 3 \right)$ 
B) 
Vertex: $\left( \frac { 11 } { 8 } , - 3 \right)$
Focus: $\left( \frac { 27 } { 8 } , - 3 \right)$ 
C) 
Vertex: $\left( \frac { 11 } { 8 } , - 3 \right)$
Focus: $\left( \frac { 75 } { 8 } , - 3 \right)$ 
D) 
Vertex: $\left( - \frac { 11 } { 8 } , 3 \right)$
Focus: $\left( \frac { 53 } { 8 } , 3 \right)$ 
E) 
Vertex: $\left( - \frac { 11 } { 8 } , - 3 \right)$
Focus: $\left( \frac { 53 } { 8 } , - 3 \right)$ 
A) 
Vertex: $\left( - \frac { 11 } { 8 } , 3 \right)$
Focus: $\left( \frac { 5 } { 8 } , 3 \right)$ 
B) 
Vertex: $\left( \frac { 11 } { 8 } , - 3 \right)$
Focus: $\left( \frac { 27 } { 8 } , - 3 \right)$ 
C) 
Vertex: $\left( \frac { 11 } { 8 } , - 3 \right)$
Focus: $\left( \frac { 75 } { 8 } , - 3 \right)$ 
D) 
Vertex: $\left( - \frac { 11 } { 8 } , 3 \right)$
Focus: $\left( \frac { 53 } { 8 } , 3 \right)$ 
E) 
Vertex: $\left( - \frac { 11 } { 8 } , - 3 \right)$
Focus: $\left( \frac { 53 } { 8 } , - 3 \right)$

Question 12

Which set of parametric equations represents the following line or conic?
UseHyperbola: 
vertices $ (5,-3),(11,-3) $
foci $ (0,-3),(16,-3) $

Accepted Answer

A)  $x = - 8 + \sqrt { 55 } \sec \theta$
$y = 3 + 3 \tan \theta$ 
B)  $x = 8 + \sqrt { 55 } \sec \theta$
$y = - 3 + 3 \tan \theta$ 
C) 
$$\begin{array} { l } 
x = - 3 + \sqrt { 55 } \sec \theta \
y = 8 + 3 \tan \theta
\end{array}$$ 
D) 
$$\begin{array} { l } 
x = 8 + 3 \sec \theta \
y = - 3 + \sqrt { 55 } \tan \theta
\end{array}$$ 
E) 
$$\begin{array} { l } 
x = - 3 + 3 \sec \theta \
y = 8 + \sqrt { 55 } \tan \theta
\end{array}$$ 
A)  $x = - 8 + \sqrt { 55 } \sec \theta$
$y = 3 + 3 \tan \theta$ 
B)  $x = 8 + \sqrt { 55 } \sec \theta$
$y = - 3 + 3 \tan \theta$ 
C) 
$$\begin{array} { l } 
x = - 3 + \sqrt { 55 } \sec \theta \
y = 8 + 3 \tan \theta
\end{array}$$ 
D) 
$$\begin{array} { l } 
x = 8 + 3 \sec \theta \
y = - 3 + \sqrt { 55 } \tan \theta
\end{array}$$ 
E) 
$$\begin{array} { l } 
x = - 3 + 3 \sec \theta \
y = 8 + \sqrt { 55 } \tan \theta
\end{array}$$

Question 13

Convert the following polar equation to rectangular form. $$\theta = - \frac { 5 \pi } { 3 }$$

Accepted Answer

A)  $y = \sqrt { 3 }$ 
B)  $y = \frac { \sqrt { 3 } } { 3 }$ 
C)  $y = - \sqrt { 3 }$ 
D)  $y = 3$ 
E)  $y = - \frac { \sqrt { 3 } } { 3 }$ 
A)  $y = \sqrt { 3 }$ 
B)  $y = \frac { \sqrt { 3 } } { 3 }$ 
C)  $y = - \sqrt { 3 }$ 
D)  $y = 3$ 
E)  $y = - \frac { \sqrt { 3 } } { 3 }$

Question 14

Use the Quadratic Formula to solve for $y$ in the following equation.
$$81 x ^ { 2 } - 108 x y + 36 y ^ { 2 } - 40 x - 50 y = 0$$

Accepted Answer

A)  $\frac { - 54 x - 25 \pm \sqrt { - 108 x + 2065 } } { 36 }$ 
B)  $\frac { 2 ( 54 x + 25 ) \pm \sqrt { - 108 x + 8260 } } { 36 }$ 
C)  $\frac { 54 x + 25 \pm \sqrt { 4140 x + 625 } } { 36 }$ 
D)  $\frac { 108 x + 50 \pm \sqrt { - 54 x + 3505 } } { 72 }$ 
E)  $\frac { - 108 x - 50 \pm \sqrt { - 54 x + 3940 } } { 72 }$ 
A)  $\frac { - 54 x - 25 \pm \sqrt { - 108 x + 2065 } } { 36 }$ 
B)  $\frac { 2 ( 54 x + 25 ) \pm \sqrt { - 108 x + 8260 } } { 36 }$ 
C)  $\frac { 54 x + 25 \pm \sqrt { 4140 x + 625 } } { 36 }$ 
D)  $\frac { 108 x + 50 \pm \sqrt { - 54 x + 3505 } } { 72 }$ 
E)  $\frac { - 108 x - 50 \pm \sqrt { - 54 x + 3940 } } { 72 }$

Question 15

Find the standard form of the equation of the ellipse below.
$2 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 0$

Accepted Answer

A)  $\frac { ( x - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } - \frac { ( y - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } - \frac { ( y + 6 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
A)  $\frac { ( x - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
B)  $\frac { ( x + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } + \frac { ( y - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$ 
C)  $\frac { ( x + 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } - \frac { ( y - 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } + \frac { ( y + 6 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } = 1$ 
E)  $\frac { ( x - 4 ) ^ { 2 } } { ( 2 \sqrt { 23 } ) ^ { 2 } } - \frac { ( y + 6 ) ^ { 2 } } { ( \sqrt { 23 } ) ^ { 2 } } = 1$

Question 16

Find a polar equation of the conic with the given characteristics and with one focus at the pole.
$\begin{array}{lcc}
\text { Conic } & \text { Eccentricity } & \text { Directrix } \
\text { Ellipse } & e=\frac{2}{5} & y=2
\end{array}$

Ellipse $e = \frac { 2 } { 5 }$ $y = 2$

Accepted Answer

A)  $r = \frac { 2 } { 5 + 2 \cos \theta }$ 
B)  $r = \frac { 4 } { 5 + 2 \cos \theta }$ 
C)  $r = \frac { 4 } { 5 - 2 \cos \theta }$ 
D)  $r = \frac { 4 } { 5 - 2 \sin \theta }$ 
E)  $r = \frac { 4 } { 5 + 2 \sin \theta }$ 
A)  $r = \frac { 2 } { 5 + 2 \cos \theta }$ 
B)  $r = \frac { 4 } { 5 + 2 \cos \theta }$ 
C)  $r = \frac { 4 } { 5 - 2 \cos \theta }$ 
D)  $r = \frac { 4 } { 5 - 2 \sin \theta }$ 
E)  $r = \frac { 4 } { 5 + 2 \sin \theta }$

Question 17

Find the standard form of the equation of the following hyperbola.
$$5 x ^ { 2 } - 3 y ^ { 2 } = 15$$

Accepted Answer

A)  $\frac { x ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } - \frac { y ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
B)  $\frac { x ^ { 2 } } { 5 ^ { 2 } } - \frac { y ^ { 2 } } { 3 ^ { 2 } } = 1$ 
C)  $\frac { y ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { x ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } = 1$ 
D)  $\frac { y ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } - \frac { x ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
E)  $\frac { x ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { y ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } = 1$ 
A)  $\frac { x ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } - \frac { y ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
B)  $\frac { x ^ { 2 } } { 5 ^ { 2 } } - \frac { y ^ { 2 } } { 3 ^ { 2 } } = 1$ 
C)  $\frac { y ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { x ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } = 1$ 
D)  $\frac { y ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } - \frac { x ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } = 1$ 
E)  $\frac { x ^ { 2 } } { ( \sqrt { 5 } ) ^ { 2 } } - \frac { y ^ { 2 } } { ( \sqrt { 3 } ) ^ { 2 } } = 1$

Question 18

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$.
$$r = \sqrt { 3 } + 2 \cos \theta$$

Accepted Answer

A)  zeros: $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
B)  zeros: $\frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 }$ 
C)  zeros: $\frac { 5 \pi } { 3 } , \frac { 7 \pi } { 3 }$ 
D)  zeros: $\frac { \pi } { 3 } , \frac { 11 \pi } { 3 }$ 
E)  zeros: $\frac { 5 \pi } { 2 } , \frac { 7 \pi } { 2 }$ 
A)  zeros: $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
B)  zeros: $\frac { 5 \pi } { 6 } , \frac { 7 \pi } { 6 }$ 
C)  zeros: $\frac { 5 \pi } { 3 } , \frac { 7 \pi } { 3 }$ 
D)  zeros: $\frac { \pi } { 3 } , \frac { 11 \pi } { 3 }$ 
E)  zeros: $\frac { 5 \pi } { 2 } , \frac { 7 \pi } { 2 }$

Question 19

Which answer is a rectangular form of the given polar equation?
$r = 18 \cos \theta$

Accepted Answer

A)  $( x - 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
B)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
C)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 9$ 
D)  $x ^ { 2 } + ( y + 9 ) ^ { 2 } = 81$ 
E)  $x ^ { 2 } + ( y - 9 ) ^ { 2 } = 81$ 
A)  $( x - 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
B)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 81$ 
C)  $( x + 9 ) ^ { 2 } + y ^ { 2 } = 9$ 
D)  $x ^ { 2 } + ( y + 9 ) ^ { 2 } = 81$ 
E)  $x ^ { 2 } + ( y - 9 ) ^ { 2 } = 81$

Question 20

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

Accepted Answer

A)  $r = \frac { 12 } { 4 - 3 \sin \theta }$ 
B)  $r = \frac { 12 } { 4 + 3 \cos \theta }$ 
C)  $r = \frac { 12 } { 4 - 3 \cos \theta }$ 
D)  $r = \frac { 12 } { 4 + 3 \sin \theta }$ 
E)  $r = \frac { 12 } { 3 - 5 \cos \theta }$ 
A)  $r = \frac { 12 } { 4 - 3 \sin \theta }$ 
B)  $r = \frac { 12 } { 4 + 3 \cos \theta }$ 
C)  $r = \frac { 12 } { 4 - 3 \cos \theta }$ 
D)  $r = \frac { 12 } { 4 + 3 \sin \theta }$ 
E)  $r = \frac { 12 } { 3 - 5 \cos \theta }$

Use the Quadratic Formula to solve for y in the following equation. $36 x ^ { 2 } - 84 x y + 49 y ^ { 2 } - 70 x - 30 y = 0$

Find the standard form of the parabola with the given characteristic and vertex at the origin. focus: $( 0 , - 5 )$

The graph of the following polar equation is a circle. Sketch its graph, and find its radius. $r = 5 \sin ( \theta )$

Solve the following system of quadratic equations algebraically by the method of substitution. $\left\{ \begin{array} { l } 8 x ^ { 2 } - 6 y ^ { 2 } + 80 = 0 \\8 x + 4 y = 0\end{array} \right.$

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: $( 0 , \pm 4 ) \quad$ foci: $( 0 , \pm 8 )$

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Ellipse e= y=5

Find the vertices and asymptotes of the hyperbola. $9 y ^ { 2 } - 4 x ^ { 2 } = 36$

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 6 } { 5 + \sin \theta }$

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

Find the vertex and focus of the parabola below. $y ^ { 2 } + 6 y - 8 x + 20 = 0$

Which set of parametric equations represents the following line or conic? UseHyperbola: vertices $(5,-3),(11,-3)$ foci $(0,-3),(16,-3)$

Convert the following polar equation to rectangular form. $\theta = - \frac { 5 \pi } { 3 }$

Use the Quadratic Formula to solve for $y$ in the following equation. $81 x ^ { 2 } - 108 x y + 36 y ^ { 2 } - 40 x - 50 y = 0$

Find the standard form of the equation of the ellipse below. $2 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 0$

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Ellipse e= y=2 Ellipse $e = \frac { 2 } { 5 }$ $y = 2$

Find the standard form of the equation of the following hyperbola. $5 x ^ { 2 } - 3 y ^ { 2 } = 15$

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$ . $r = \sqrt { 3 } + 2 \cos \theta$

Which answer is a rectangular form of the given polar equation? $r = 18 \cos \theta$

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Sequences, Series, and Probability

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Exam 9: Topics in Analytic Geometry

Use the Quadratic Formula to solve for y in the following equation. 36x2−84xy+49y2−70x−30y=036 x ^ { 2 } - 84 x y + 49 y ^ { 2 } - 70 x - 30 y = 036x2−84xy+49y2−70x−30y=0

Find the standard form of the parabola with the given characteristic and vertex at the origin. focus: (0,−5)( 0 , - 5 )(0,−5)

The graph of the following polar equation is a circle. Sketch its graph, and find its radius. r=5sin⁡(θ)r = 5 \sin ( \theta )r=5sin(θ)

Solve the following system of quadratic equations algebraically by the method of substitution. {8x2−6y2+80=08x+4y=0\left\{ \begin{array} { l } 8 x ^ { 2 } - 6 y ^ { 2 } + 80 = 0 \\8 x + 4 y = 0\end{array} \right.{8x2−6y2+80=08x+4y=0​

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (0,±4)( 0 , \pm 4 ) \quad(0,±4) foci: (0,±8)( 0 , \pm 8 )(0,±8)

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Ellipse e= y=5

Find the vertices and asymptotes of the hyperbola. 9y2−4x2=369 y ^ { 2 } - 4 x ^ { 2 } = 369y2−4x2=36

Sketch the graph of the polar equation using symmetry, zeros, maximum rrr -values, and any other additional points. r2=4sin⁡2θr ^ { 2 } = 4 \sin 2 \thetar2=4sin2θ Use either grid below for your graph, whichever is more convenient.

Test for symmetry with respect to θ=π/2\theta = \pi / 2θ=π/2 , the polar axis, and the pole. r=65+sin⁡θr = \frac { 6 } { 5 + \sin \theta }r=5+sinθ6​

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

Find the vertex and focus of the parabola below. y2+6y−8x+20=0y ^ { 2 } + 6 y - 8 x + 20 = 0y2+6y−8x+20=0

Which set of parametric equations represents the following line or conic? UseHyperbola: vertices (5,−3),(11,−3) (5,-3),(11,-3) (5,−3),(11,−3) foci (0,−3),(16,−3) (0,-3),(16,-3) (0,−3),(16,−3)

Convert the following polar equation to rectangular form. θ=−5π3\theta = - \frac { 5 \pi } { 3 }θ=−35π​

Use the Quadratic Formula to solve for yyy in the following equation. 81x2−108xy+36y2−40x−50y=081 x ^ { 2 } - 108 x y + 36 y ^ { 2 } - 40 x - 50 y = 081x2−108xy+36y2−40x−50y=0

Find the standard form of the equation of the ellipse below. 2x2+8y2+24x−64y+16=02 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 02x2+8y2+24x−64y+16=0

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Ellipse e= y=2 Ellipse e=25e = \frac { 2 } { 5 }e=52​ y=2y = 2y=2

Find the standard form of the equation of the following hyperbola. 5x2−3y2=155 x ^ { 2 } - 3 y ^ { 2 } = 155x2−3y2=15

Find any zeros of rrr on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π . r=3+2cos⁡θr = \sqrt { 3 } + 2 \cos \thetar=3​+2cosθ

Which answer is a rectangular form of the given polar equation? r=18cos⁡θr = 18 \cos \thetar=18cosθ

Find a polar equation of the conic with the given characteristics and with one focus at the pole.

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

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Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

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Use the Quadratic Formula to solve for y in the following equation. $36 x ^ { 2 } - 84 x y + 49 y ^ { 2 } - 70 x - 30 y = 0$

Find the standard form of the parabola with the given characteristic and vertex at the origin. focus: $( 0 , - 5 )$

The graph of the following polar equation is a circle. Sketch its graph, and find its radius. $r = 5 \sin ( \theta )$

Solve the following system of quadratic equations algebraically by the method of substitution. $\left\{ \begin{array} { l } 8 x ^ { 2 } - 6 y ^ { 2 } + 80 = 0 \\8 x + 4 y = 0\end{array} \right.$

Find the standard form of the equation of the hyperbola with the given characteristics. vertices: $( 0 , \pm 4 ) \quad$ foci: $( 0 , \pm 8 )$

Find the vertices and asymptotes of the hyperbola. $9 y ^ { 2 } - 4 x ^ { 2 } = 36$

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 6 } { 5 + \sin \theta }$

Find the vertex and focus of the parabola below. $y ^ { 2 } + 6 y - 8 x + 20 = 0$

Which set of parametric equations represents the following line or conic? UseHyperbola: vertices $(5,-3),(11,-3)$ foci $(0,-3),(16,-3)$

Convert the following polar equation to rectangular form. $\theta = - \frac { 5 \pi } { 3 }$

Use the Quadratic Formula to solve for $y$ in the following equation. $81 x ^ { 2 } - 108 x y + 36 y ^ { 2 } - 40 x - 50 y = 0$

Find the standard form of the equation of the ellipse below. $2 x ^ { 2 } + 8 y ^ { 2 } + 24 x - 64 y + 16 = 0$

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Ellipse e= y=2 Ellipse $e = \frac { 2 } { 5 }$ $y = 2$

Find the standard form of the equation of the following hyperbola. $5 x ^ { 2 } - 3 y ^ { 2 } = 15$

Find any zeros of $r$ on the interval $0 \leq \theta < 2 \pi$ . $r = \sqrt { 3 } + 2 \cos \theta$

Which answer is a rectangular form of the given polar equation? $r = 18 \cos \theta$