Question 1

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = 5 + 4 \cos ( \theta )$$

Accepted Answer

A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries. 
A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries.

Question 2

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 3

Find the standard form of the equation of the ellipse centered at the origin having vertices at $( - 4,0 )$ and $( 4,0 )$ and foci at $( - 1,0 )$ and $( 1,0 )$.

Accepted Answer

A)  $\frac { x ^ { 2 } } { ( \sqrt { 15 } ) ^ { 2 } } + \frac { y ^ { 2 } } { 4 ^ { 2 } } = 1$ 
B)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { \sqrt { 15 } } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { \sqrt { 17 } } = 1$ 
D)  $\frac { x ^ { 2 } } { \sqrt { 15 } } + \frac { y ^ { 2 } } { 4 } = 1$ 
E)  $\frac { x ^ { 2 } } { 4 ^ { 2 } } + \frac { y ^ { 2 } } { ( \sqrt { 15 } ) ^ { 2 } } = 1$ 
A)  $\frac { x ^ { 2 } } { ( \sqrt { 15 } ) ^ { 2 } } + \frac { y ^ { 2 } } { 4 ^ { 2 } } = 1$ 
B)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { \sqrt { 15 } } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { \sqrt { 17 } } = 1$ 
D)  $\frac { x ^ { 2 } } { \sqrt { 15 } } + \frac { y ^ { 2 } } { 4 } = 1$ 
E)  $\frac { x ^ { 2 } } { 4 ^ { 2 } } + \frac { y ^ { 2 } } { ( \sqrt { 15 } ) ^ { 2 } } = 1$

Question 4

Use the discriminant to classify the graph; then use the quadratic formula to solve for $y$.
$$3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } - 18 \sqrt { 3 } x + 30 y + 36 = 0$$

Accepted Answer

A)  ellipse; $\quad y = - 15 + \sqrt { 3 } x \pm \sqrt { 12 \sqrt { 3 } x + 189 }$ 
B)  ellipse; $y = - 15 + \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$ 
C)  parabola; $y = - 15 + \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$ 
D)  parabola; $y = \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$ 
E)  hyperbola; $y = \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$ 
A)  ellipse; $\quad y = - 15 + \sqrt { 3 } x \pm \sqrt { 12 \sqrt { 3 } x + 189 }$ 
B)  ellipse; $y = - 15 + \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$ 
C)  parabola; $y = - 15 + \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$ 
D)  parabola; $y = \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$ 
E)  hyperbola; $y = \sqrt { 3 } x \pm \sqrt { - 12 \sqrt { 3 } x + 189 }$

Question 5

Which answer is a polar form of the given rectangular equation? $25 x y = 225$

Accepted Answer

A)  $r ^ { 2 } = 9 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 3 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 3 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 9 \sin \theta \cos \theta$ 
E)  $\theta = 9 \sin \theta \cos \theta$ 
A)  $r ^ { 2 } = 9 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 3 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 3 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 9 \sin \theta \cos \theta$ 
E)  $\theta = 9 \sin \theta \cos \theta$

Question 6

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

Accepted Answer

A)  zeros: $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  zeros: $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  zeros: $\frac { \pi } { 2 } , \frac { 7 \pi } { 2 }$ 
D)  zeros: $\frac { 3 \pi } { 2 } , \frac { 5 \pi } { 2 }$ 
E)  zeros: $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 }$ 
A)  zeros: $\frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 }$ 
B)  zeros: $\frac { \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  zeros: $\frac { \pi } { 2 } , \frac { 7 \pi } { 2 }$ 
D)  zeros: $\frac { 3 \pi } { 2 } , \frac { 5 \pi } { 2 }$ 
E)  zeros: $\frac { \pi } { 8 } , \frac { 7 \pi } { 8 }$

Question 7

Find the center and vertices of the ellipse.
$$x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0$$

Accepted Answer

A)  center: $( 8,1 ) \quad$ vertices: $( 4,1 ) , ( 12,1 )$ 
B)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 2 , - 8 ) , ( 0 , - 8 )$ 
C)  center: $( 1,8 ) \quad$ vertices: $( 0,8 ) , ( 2,8 )$ 
D)  center: $( 1,8 )$ vertices: $( - 3,8 ) , ( 5,8 )$ 
E)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 5 , - 8 ) , ( 3 , - 8 )$ 
A)  center: $( 8,1 ) \quad$ vertices: $( 4,1 ) , ( 12,1 )$ 
B)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 2 , - 8 ) , ( 0 , - 8 )$ 
C)  center: $( 1,8 ) \quad$ vertices: $( 0,8 ) , ( 2,8 )$ 
D)  center: $( 1,8 )$ vertices: $( - 3,8 ) , ( 5,8 )$ 
E)  center: $( - 1 , - 8 ) \quad$ vertices: $( - 5 , - 8 ) , ( 3 , - 8 )$

Question 8

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$$64 x ^ { 2 } + 64 y ^ { 2 } + 192 x + 24 y - 327 = 0$$

Accepted Answer

A)  hyperbola 
B)  ellipse 
C)  circle 
D)  parabola 
A)  hyperbola 
B)  ellipse 
C)  circle 
D)  parabola

Question 9

Match the graph with its equation.

Accepted Answer

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

Question 10

Find the standard form of the parabola with the given characteristic and vertex at the origin.
vertical axis and passes through point $( 10 , - 10 )$

Accepted Answer

A)  $y ^ { 2 } = - 10 x$ 
B)  $x ^ { 2 } = - 10 y$ 
C)  $x ^ { 2 } = \frac { 5 } { 2 } y$ 
D)  $y ^ { 2 } = \frac { 5 } { 2 } x$ 
E)  $y ^ { 2 } = \frac { 5 } { 2 } x ^ { 2 }$ 
A)  $y ^ { 2 } = - 10 x$ 
B)  $x ^ { 2 } = - 10 y$ 
C)  $x ^ { 2 } = \frac { 5 } { 2 } y$ 
D)  $y ^ { 2 } = \frac { 5 } { 2 } x$ 
E)  $y ^ { 2 } = \frac { 5 } { 2 } x ^ { 2 }$

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

Rotate the axes to eliminate the $x y$-term in the equation. Then write the equation in standard form.
$41 x ^ { 2 } + 18 x y + 41 y ^ { 2 } - 800 = 0$

Accepted Answer

A)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 25 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 36 } = 1$ 
B)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 9 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 36 } = 1$ 
D)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 9 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 25 } = 1$ 
E)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 25 } = 1$ 
A)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 25 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 36 } = 1$ 
B)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 9 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 36 } = 1$ 
D)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 9 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 25 } = 1$ 
E)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 16 } + \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 25 } = 1$

Question 13

Find the center and foci of the hyperbola.
$$\frac { ( y + 1 ) ^ { 2 } } { 25 } - \frac { ( x + 2 ) ^ { 2 } } { 24 } = 1$$

Accepted Answer

A)  center: $( 2,1 )$ foci: $( 2 , - 6 ) , ( 2,8 )$ 
B)  center: $( - 2 , - 1 ) \quad$ foci: $( - 2 , - 8 ) , ( - 2,6 )$ 
C)  center: $( - 1 , - 2 ) \quad$ foci: $( - 8 , - 2 ) , ( 6 , - 2 )$ 
D)  center: $( - 2 , - 1 ) \quad$ foci: $( - 9 , - 1 ) , ( 5 , - 1 )$ 
E)  center: $( 1,2 )$
foci: $( 1 , - 5 ) , ( 1,9 )$ 
A)  center: $( 2,1 )$ foci: $( 2 , - 6 ) , ( 2,8 )$ 
B)  center: $( - 2 , - 1 ) \quad$ foci: $( - 2 , - 8 ) , ( - 2,6 )$ 
C)  center: $( - 1 , - 2 ) \quad$ foci: $( - 8 , - 2 ) , ( 6 , - 2 )$ 
D)  center: $( - 2 , - 1 ) \quad$ foci: $( - 9 , - 1 ) , ( 5 , - 1 )$ 
E)  center: $( 1,2 )$
foci: $( 1 , - 5 ) , ( 1,9 )$

Question 14

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = - 6 + \cos ( 5 \theta )$$

Accepted Answer

A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries. 
A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the polar axis only. 
C)  The graph is symmetric with respect to the pole only. 
D)  The graph is symmetric with respect to all three. 
E)  The graph has no symmetries.

Question 15

Find a polar equation of the conic with the given characteristics and with one focus at the pole.
$\begin{array} { c c c } \text { Conic } & \text { Eccentricity } & \text { Directrix } \ \text { Parabola } & e = 1 & x = - 4 \end{array}$

Accepted Answer

A)  $r = \frac { 4 } { 1 + \sin \theta }$ 
B)  $r = \frac { 4 } { 1 - \sin \theta }$ 
C)  $r = \frac { 4 } { 1 - \cos \theta }$ 
D)  $r = \frac { 4 } { 1 + \cos \theta }$ 
E)  $r = \frac { 1 } { 1 - 4 \cos \theta }$ 
A)  $r = \frac { 4 } { 1 + \sin \theta }$ 
B)  $r = \frac { 4 } { 1 - \sin \theta }$ 
C)  $r = \frac { 4 } { 1 - \cos \theta }$ 
D)  $r = \frac { 4 } { 1 + \cos \theta }$ 
E)  $r = \frac { 1 } { 1 - 4 \cos \theta }$

Question 16

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 17

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$, the polar axis, and the pole.
$$r = - 2 + \sin ( 2 \theta )$$

Accepted Answer

A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the pole only. 
C)  The graph is symmetric with respect to all three. 
D)  The graph has no symmetries. 
E)  The graph is symmetric with respect to the polar axis only. 
A)  The graph is symmetric with respect to $\theta = \frac { \pi } { 2 }$ only. 
B)  The graph is symmetric with respect to the pole only. 
C)  The graph is symmetric with respect to all three. 
D)  The graph has no symmetries. 
E)  The graph is symmetric with respect to the polar axis only.

Question 18

Which set of parametric equations represents the following line or conic?
Use $x = h + a \sec \theta$ and $y = k + b \tan \theta$.
Hyperbola: vertices $( - 5,7 ) , ( - 3,7 )$
foci $( - 8,7 ) , ( 0,7 )$

Accepted Answer

A) 
$\begin{array} { l } 
x = 4 + \sqrt { 15 } \sec \theta \
y = - 7 + \tan \theta
\end{array}$ 
B) 
$\begin{array} { l } 
x = - 4 + \sqrt { 15 } \sec \theta \
y = 7 + \tan \theta
\end{array}$ 
C) 
$\begin{array} { l } 
x = 7 + \sqrt { 15 } \sec \theta \
y = - 4 + \tan \theta
\end{array}$ 
D) 
$x = - 4 + \sec \theta$
$y = 7 + \sqrt { 15 } \tan \theta$ 
E) 
$\begin{array} { l } 
x = 7 + \sec \theta \
y = - 4 + \sqrt { 15 } \tan \theta
\end{array}$ 
A) 
$\begin{array} { l } 
x = 4 + \sqrt { 15 } \sec \theta \
y = - 7 + \tan \theta
\end{array}$ 
B) 
$\begin{array} { l } 
x = - 4 + \sqrt { 15 } \sec \theta \
y = 7 + \tan \theta
\end{array}$ 
C) 
$\begin{array} { l } 
x = 7 + \sqrt { 15 } \sec \theta \
y = - 4 + \tan \theta
\end{array}$ 
D) 
$x = - 4 + \sec \theta$
$y = 7 + \sqrt { 15 } \tan \theta$ 
E) 
$\begin{array} { l } 
x = 7 + \sec \theta \
y = - 4 + \sqrt { 15 } \tan \theta
\end{array}$

Question 19

Rotate the axes to eliminate the $x y$-term in the following equation.
$$- 30 x ^ { 2 } + 20 \sqrt { 3 } x y - 10 y ^ { 2 } - 20 x - 20 \sqrt { 3 } y = 0$$

Accepted Answer

A)  $\left( y ^ { \prime } \right) ^ { 2 } + x ^ { \prime } = 0$ 
B)  $- 10 \sqrt { 3 } \left( y ^ { \prime } \right) ^ { 2 } - 10 \sqrt { 3 } x ^ { \prime } = 0$ 
C)  $- 10 \left( x ^ { \prime } \right) ^ { 2 } + 10 y ^ { \prime } = 0$ 
D)  $- \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 10 } - \frac { x ^ { \prime } } { 10 } = 0$ 
E)  $- \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 20 } - \frac { y ^ { \prime } } { 20 } = 0$ 
A)  $\left( y ^ { \prime } \right) ^ { 2 } + x ^ { \prime } = 0$ 
B)  $- 10 \sqrt { 3 } \left( y ^ { \prime } \right) ^ { 2 } - 10 \sqrt { 3 } x ^ { \prime } = 0$ 
C)  $- 10 \left( x ^ { \prime } \right) ^ { 2 } + 10 y ^ { \prime } = 0$ 
D)  $- \frac { \left( y ^ { \prime } \right) ^ { 2 } } { 10 } - \frac { x ^ { \prime } } { 10 } = 0$ 
E)  $- \frac { \left( x ^ { \prime } \right) ^ { 2 } } { 20 } - \frac { y ^ { \prime } } { 20 } = 0$

Question 20

Rotate the axes to eliminate the $x y$-term in the equation below and then write the equation in standard form.
$$x y + 1 = 0$$

Accepted Answer

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

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = 5 + 4 \cos ( \theta )$

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 centered at the origin having vertices at $( - 4,0 )$ and $( 4,0 )$ and foci at $( - 1,0 )$ and $( 1,0 )$ .

Use the discriminant to classify the graph; then use the quadratic formula to solve for $y$ . $3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } - 18 \sqrt { 3 } x + 30 y + 36 = 0$

Which answer is a polar form of the given rectangular equation? $25 x y = 225$

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

Find the center and vertices of the ellipse. $x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $64 x ^ { 2 } + 64 y ^ { 2 } + 192 x + 24 y - 327 = 0$

Match the graph with its equation.

Find the standard form of the parabola with the given characteristic and vertex at the origin. vertical axis and passes through point $( 10 , - 10 )$

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

Rotate the axes to eliminate the $x y$ -term in the equation. Then write the equation in standard form. $41 x ^ { 2 } + 18 x y + 41 y ^ { 2 } - 800 = 0$

Find the center and foci of the hyperbola. $\frac { ( y + 1 ) ^ { 2 } } { 25 } - \frac { ( x + 2 ) ^ { 2 } } { 24 } = 1$

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = - 6 + \cos ( 5 \theta )$

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Parabola e=1 x=-4

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

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = - 2 + \sin ( 2 \theta )$

Which set of parametric equations represents the following line or conic? Use $x = h + a \sec \theta$ and $y = k + b \tan \theta$ . Hyperbola: vertices $( - 5,7 ) , ( - 3,7 )$ foci $( - 8,7 ) , ( 0,7 )$

Rotate the axes to eliminate the $x y$ -term in the following equation. $- 30 x ^ { 2 } + 20 \sqrt { 3 } x y - 10 y ^ { 2 } - 20 x - 20 \sqrt { 3 } y = 0$

Rotate the axes to eliminate the $x y$ -term in the equation below and then write the equation in standard form. $x y + 1 = 0$

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Exam 9: Topics in Analytic Geometry

Test the graph of the following equation for symmetry with respect to θ=π2\theta = \frac { \pi } { 2 }θ=2π​ , the polar axis, and the pole. r=5+4cos⁡(θ)r = 5 + 4 \cos ( \theta )r=5+4cos(θ)

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 centered at the origin having vertices at (−4,0)( - 4,0 )(−4,0) and (4,0)( 4,0 )(4,0) and foci at (−1,0)( - 1,0 )(−1,0) and (1,0)( 1,0 )(1,0) .

Use the discriminant to classify the graph; then use the quadratic formula to solve for yyy . 3x2−23xy+y2−183x+30y+36=03 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } - 18 \sqrt { 3 } x + 30 y + 36 = 03x2−23​xy+y2−183​x+30y+36=0

Which answer is a polar form of the given rectangular equation? 25xy=22525 x y = 22525xy=225

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

Find the center and vertices of the ellipse. x2+16y2−2x−256y+1009=0x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0x2+16y2−2x−256y+1009=0

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 64x2+64y2+192x+24y−327=064 x ^ { 2 } + 64 y ^ { 2 } + 192 x + 24 y - 327 = 064x2+64y2+192x+24y−327=0

Match the graph with its equation.

Find the standard form of the parabola with the given characteristic and vertex at the origin. vertical axis and passes through point (10,−10)( 10 , - 10 )(10,−10)

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

Rotate the axes to eliminate the xyx yxy -term in the equation. Then write the equation in standard form. 41x2+18xy+41y2−800=041 x ^ { 2 } + 18 x y + 41 y ^ { 2 } - 800 = 041x2+18xy+41y2−800=0

Find the center and foci of the hyperbola. (y+1)225−(x+2)224=1\frac { ( y + 1 ) ^ { 2 } } { 25 } - \frac { ( x + 2 ) ^ { 2 } } { 24 } = 125(y+1)2​−24(x+2)2​=1

Test the graph of the following equation for symmetry with respect to θ=π2\theta = \frac { \pi } { 2 }θ=2π​ , the polar axis, and the pole. r=−6+cos⁡(5θ)r = - 6 + \cos ( 5 \theta )r=−6+cos(5θ)

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Parabola e=1 x=-4

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

Test the graph of the following equation for symmetry with respect to θ=π2\theta = \frac { \pi } { 2 }θ=2π​ , the polar axis, and the pole. r=−2+sin⁡(2θ)r = - 2 + \sin ( 2 \theta )r=−2+sin(2θ)

Rotate the axes to eliminate the xyx yxy -term in the following equation. −30x2+203xy−10y2−20x−203y=0- 30 x ^ { 2 } + 20 \sqrt { 3 } x y - 10 y ^ { 2 } - 20 x - 20 \sqrt { 3 } y = 0−30x2+203​xy−10y2−20x−203​y=0

Rotate the axes to eliminate the xyx yxy -term in the equation below and then write the equation in standard form. xy+1=0x y + 1 = 0xy+1=0

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

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Sequences, Series, and Probability

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

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Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = 5 + 4 \cos ( \theta )$

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 centered at the origin having vertices at $( - 4,0 )$ and $( 4,0 )$ and foci at $( - 1,0 )$ and $( 1,0 )$ .

Use the discriminant to classify the graph; then use the quadratic formula to solve for $y$ . $3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } - 18 \sqrt { 3 } x + 30 y + 36 = 0$

Which answer is a polar form of the given rectangular equation? $25 x y = 225$

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

Find the center and vertices of the ellipse. $x ^ { 2 } + 16 y ^ { 2 } - 2 x - 256 y + 1009 = 0$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $64 x ^ { 2 } + 64 y ^ { 2 } + 192 x + 24 y - 327 = 0$

Find the standard form of the parabola with the given characteristic and vertex at the origin. vertical axis and passes through point $( 10 , - 10 )$

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

Rotate the axes to eliminate the $x y$ -term in the equation. Then write the equation in standard form. $41 x ^ { 2 } + 18 x y + 41 y ^ { 2 } - 800 = 0$

Find the center and foci of the hyperbola. $\frac { ( y + 1 ) ^ { 2 } } { 25 } - \frac { ( x + 2 ) ^ { 2 } } { 24 } = 1$

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = - 6 + \cos ( 5 \theta )$

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

Test the graph of the following equation for symmetry with respect to $\theta = \frac { \pi } { 2 }$ , the polar axis, and the pole. $r = - 2 + \sin ( 2 \theta )$

Rotate the axes to eliminate the $x y$ -term in the following equation. $- 30 x ^ { 2 } + 20 \sqrt { 3 } x y - 10 y ^ { 2 } - 20 x - 20 \sqrt { 3 } y = 0$

Rotate the axes to eliminate the $x y$ -term in the equation below and then write the equation in standard form. $x y + 1 = 0$