Question 1

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: $y = \pm \frac { 2 } { 3 } x$ A

Question 2

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$$y ^ { 2 } + 13 x + y - 79 = 0$$

Accepted Answer

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

Question 3

Use the Quadratic Formula to solve for $y$ in the following equation.

Accepted Answer

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

Question 4

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 5

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ?
$$y = x ^ { 2 } + 9$$

Accepted Answer

A) 
$\begin{array} { l } 
x = t + 6 \
y = ( t + 6 ) ^ { 2 } - 9
\end{array}$ 
B) 
$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } - 9
\end{array}$ 
C) 
$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } + 9
\end{array}$ 
D) 
$\begin{array} { l } 
x = 6 + t \
y = ( 6 + t ) ^ { 2 } + 9
\end{array}$ 
E) 
$\begin{array} { l } 
x = 6 - t \
y = ( 6 - t ) ^ { 2 } + 9
\end{array}$ 
A) 
$\begin{array} { l } 
x = t + 6 \
y = ( t + 6 ) ^ { 2 } - 9
\end{array}$ 
B) 
$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } - 9
\end{array}$ 
C) 
$\begin{array} { l } 
x = t - 6 \
y = ( t - 6 ) ^ { 2 } + 9
\end{array}$ 
D) 
$\begin{array} { l } 
x = 6 + t \
y = ( 6 + t ) ^ { 2 } + 9
\end{array}$ 
E) 
$\begin{array} { l } 
x = 6 - t \
y = ( 6 - t ) ^ { 2 } + 9
\end{array}$

Question 6

Sketch the graph of the polar equation using symmetry, zeros, maximum $r$-values, and any other additional points.
$$r = 2 - 3 \cos \theta$$
Use either grid below for your graph, whichever is more convenient.

Accepted Answer

symmetry: with respe

Question 7

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 8

Rotate the axes to eliminate the $x y$-term in the following equation and then write the equation in standard form.
$$3 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 0$$

Accepted Answer

A) 
$\left( x ^ { \prime } - 1 \right) ^ { 2 } = 3 \left( y ^ { \prime } - \frac { 4 } { 5 } \right)$ 
B)  $\left( x ^ { \prime } + 3 \right) ^ { 2 } = - 5 \left( y ^ { \prime } + \frac { 1 } { 3 } \right)$ 
C)  $\left( y ^ { \prime } + 1 \right) ^ { 2 } = - \left( x ^ { \prime } - \frac { 4 } { 5 } \right)$ 
D)  $\left( y ^ { \prime } - \frac { 4 } { 5 } \right) ^ { 2 } = 3 \left( x ^ { \prime } - 1 \right)$ 
E)  $\left( y ^ { \prime } + \frac { 4 } { 5 } \right) ^ { 2 } = x ^ { \prime } + \frac { 1 } { 3 }$ 
A) 
$\left( x ^ { \prime } - 1 \right) ^ { 2 } = 3 \left( y ^ { \prime } - \frac { 4 } { 5 } \right)$ 
B)  $\left( x ^ { \prime } + 3 \right) ^ { 2 } = - 5 \left( y ^ { \prime } + \frac { 1 } { 3 } \right)$ 
C)  $\left( y ^ { \prime } + 1 \right) ^ { 2 } = - \left( x ^ { \prime } - \frac { 4 } { 5 } \right)$ 
D)  $\left( y ^ { \prime } - \frac { 4 } { 5 } \right) ^ { 2 } = 3 \left( x ^ { \prime } - 1 \right)$ 
E)  $\left( y ^ { \prime } + \frac { 4 } { 5 } \right) ^ { 2 } = x ^ { \prime } + \frac { 1 } { 3 }$

Question 9

Test for symmetry with respect to $\theta = \pi / 2$, the polar axis, and the pole.
$$r = \frac { 2 } { 4 + \sin \theta }$$

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 the standard form of the parabola with the given characteristics.
focus: $( 8,13 ) \quad$ directrix: $y = - 1$

Accepted Answer

A)  $( x - 8 ) ^ { 2 } = 28 ( y - 6 )$ 
B)  $( x + 8 ) ^ { 2 } = 28 ( y + 6 )$ 
C)  $( x - 6 ) ^ { 2 } = 28 ( y - 8 )$ 
D)  $( x + 6 ) ^ { 2 } = 28 ( y + 8 )$ 
E)  $( x + 8 ) ^ { 2 } = 7 ( y + 6 )$ 
A)  $( x - 8 ) ^ { 2 } = 28 ( y - 6 )$ 
B)  $( x + 8 ) ^ { 2 } = 28 ( y + 6 )$ 
C)  $( x - 6 ) ^ { 2 } = 28 ( y - 8 )$ 
D)  $( x + 6 ) ^ { 2 } = 28 ( y + 8 )$ 
E)  $( x + 8 ) ^ { 2 } = 7 ( y + 6 )$

Question 11

Find the graph of the following polar equation.
$$r = 6 \sin ( 2 \theta )$$

Accepted Answer

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

Question 12

Rotate the axes to eliminate the $x y$-term in the following equation.
$$18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$$

Accepted Answer

A)  $\left( y ^ { \prime } \right) ^ { 2 } + x ^ { \prime } = 0$ 
B)  $6 \sqrt { 3 } \left( y ^ { \prime } \right) ^ { 2 } + 6 \sqrt { 3 } x ^ { \prime } = 0$ 
C)  $6 \left( x ^ { \prime } \right) ^ { 2 } - 6 y ^ { \prime } = 0$ 
D)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { x ^ { \prime } } { 6 } = 0$ 
E)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 12 } + \frac { y ^ { \prime } } { 12 } = 0$ 
A)  $\left( y ^ { \prime } \right) ^ { 2 } + x ^ { \prime } = 0$ 
B)  $6 \sqrt { 3 } \left( y ^ { \prime } \right) ^ { 2 } + 6 \sqrt { 3 } x ^ { \prime } = 0$ 
C)  $6 \left( x ^ { \prime } \right) ^ { 2 } - 6 y ^ { \prime } = 0$ 
D)  $\frac { \left( y ^ { \prime } \right) ^ { 2 } } { 6 } + \frac { x ^ { \prime } } { 6 } = 0$ 
E)  $\frac { \left( x ^ { \prime } \right) ^ { 2 } } { 12 } + \frac { y ^ { \prime } } { 12 } = 0$

Question 13

Convert the point from polar coordinates to rectangular coordinates. Round answer to three decimal places, if necessary.
$$\left( 3 , - \frac { 3 \pi } { 2 } \right)$$

Accepted Answer

A)  $( 3,0 )$ 
B)  $( 0,3 )$ 
C)  $( 0 , - 3 )$ 
D)  $( - 3,0 )$ 
E)  $( 3 , - 4.712 )$ 
A)  $( 3,0 )$ 
B)  $( 0,3 )$ 
C)  $( 0 , - 3 )$ 
D)  $( - 3,0 )$ 
E)  $( 3 , - 4.712 )$

Question 14

A projectile is launched from ground level 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
$x = \left( v _ { 0 } \cos \theta \right) t$ and $y = \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 }$.
Use a graphing utility to graph the paths of a projectile launched from ground level with the values given for $\theta$ and $v _ { 0 }$. Use the graph to approximate the maximum height and range of the projectile to the nearest foot.
$\theta = 50 ^ { \circ } , \quad v _ { 0 } = 104$ feet per second

Accepted Answer

A)  maximum height $\approx 12$ feet range $\approx 171$ feet 
B)  maximum height $\approx 171$ feet range $\approx 12$ feet 
C)  maximum height $\approx 99$ feet range $\approx 333$ feet 
D)  maximum height $\approx 333$ feet range $\approx 99$ feet 
E)  maximum height $\approx 38$ feet range $\approx 8$ feet 
A)  maximum height $\approx 12$ feet range $\approx 171$ feet 
B)  maximum height $\approx 171$ feet range $\approx 12$ feet 
C)  maximum height $\approx 99$ feet range $\approx 333$ feet 
D)  maximum height $\approx 333$ feet range $\approx 99$ feet 
E)  maximum height $\approx 38$ feet range $\approx 8$ feet

Question 15

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 16

Find the standard form of the equation of the ellipse with the given characteristics.
vertices: $( 4 , - 6 ) , ( 4,10 ) \quad$ minor axis of length: 4

Accepted Answer

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

Question 17

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 $\quad ( 5 , - 3 ) , ( 11 , - 3 )$
foci $( 0 , - 3 ) , ( 16 , - 3 )$

Accepted Answer

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

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 20

Find the standard form of the parabola with the given characteristic and vertex at the origin.
directrix: $x = 5$

Accepted Answer

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

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 13 x + y - 79 = 0$

Use the Quadratic Formula to solve for $y$ in the following equation.

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

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 9$

Match the graph with its equation.

Rotate the axes to eliminate the $x y$ -term in the following equation and then write the equation in standard form. $3 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 0$

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 2 } { 4 + \sin \theta }$

Find the standard form of the parabola with the given characteristics. focus: $( 8,13 ) \quad$ directrix: $y = - 1$

Find the graph of the following polar equation. $r = 6 \sin ( 2 \theta )$

Rotate the axes to eliminate the $x y$ -term in the following equation. $18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$

Convert the point from polar coordinates to rectangular coordinates. Round answer to three decimal places, if necessary. $\left( 3 , - \frac { 3 \pi } { 2 } \right)$

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

Find the standard form of the equation of the ellipse with the given characteristics. vertices: $( 4 , - 6 ) , ( 4,10 ) \quad$ minor axis of length: 4

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 $\quad ( 5 , - 3 ) , ( 11 , - 3 )$ foci $( 0 , - 3 ) , ( 16 , - 3 )$

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

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 parabola with the given characteristic and vertex at the origin. directrix: $x = 5$

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

Calculus Practice Problems

Filters

Exam 9: Topics in Analytic Geometry

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. y2+13x+y−79=0y ^ { 2 } + 13 x + y - 79 = 0y2+13x+y−79=0

Use the Quadratic Formula to solve for yyy in the following equation.

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 graph of the following rectangular equation using t=6−xt = 6 - xt=6−x ? y=x2+9y = x ^ { 2 } + 9y=x2+9

Sketch the graph of the polar equation using symmetry, zeros, maximum rrr -values, and any other additional points. r=2−3cos⁡θr = 2 - 3 \cos \thetar=2−3cosθ Use either grid below for your graph, whichever is more convenient.

Match the graph with its equation.

Rotate the axes to eliminate the xyx yxy -term in the following equation and then write the equation in standard form. 3x2−12xy+12y2+155y+3=03 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 03x2−12xy+12y2+155​y+3=0

Test for symmetry with respect to θ=π/2\theta = \pi / 2θ=π/2 , the polar axis, and the pole. r=24+sin⁡θr = \frac { 2 } { 4 + \sin \theta }r=4+sinθ2​

Find the standard form of the parabola with the given characteristics. focus: (8,13)( 8,13 ) \quad(8,13) directrix: y=−1y = - 1y=−1

Find the graph of the following polar equation. r=6sin⁡(2θ)r = 6 \sin ( 2 \theta )r=6sin(2θ)

Rotate the axes to eliminate the xyx yxy -term in the following equation. 18x2−123xy+6y2+12x+123y=018 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 018x2−123​xy+6y2+12x+123​y=0

Convert the point from polar coordinates to rectangular coordinates. Round answer to three decimal places, if necessary. (3,−3π2)\left( 3 , - \frac { 3 \pi } { 2 } \right)(3,−23π​)

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

Find the standard form of the equation of the ellipse with the given characteristics. vertices: (4,−6),(4,10)( 4 , - 6 ) , ( 4,10 ) \quad(4,−6),(4,10) minor axis of length: 4

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

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 the standard form of the parabola with the given characteristic and vertex at the origin. directrix: x=5x = 5x=5

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

Calculus Practice Problems

Filters

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $y ^ { 2 } + 13 x + y - 79 = 0$

Use the Quadratic Formula to solve for $y$ in the following equation.

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

Which set of parametric equations represents the graph of the following rectangular equation using $t = 6 - x$ ? $y = x ^ { 2 } + 9$

Rotate the axes to eliminate the $x y$ -term in the following equation and then write the equation in standard form. $3 x ^ { 2 } - 12 x y + 12 y ^ { 2 } + 15 \sqrt { 5 } y + 3 = 0$

Test for symmetry with respect to $\theta = \pi / 2$ , the polar axis, and the pole. $r = \frac { 2 } { 4 + \sin \theta }$

Find the standard form of the parabola with the given characteristics. focus: $( 8,13 ) \quad$ directrix: $y = - 1$

Find the graph of the following polar equation. $r = 6 \sin ( 2 \theta )$

Rotate the axes to eliminate the $x y$ -term in the following equation. $18 x ^ { 2 } - 12 \sqrt { 3 } x y + 6 y ^ { 2 } + 12 x + 12 \sqrt { 3 } y = 0$

Convert the point from polar coordinates to rectangular coordinates. Round answer to three decimal places, if necessary. $\left( 3 , - \frac { 3 \pi } { 2 } \right)$

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

Find the standard form of the equation of the ellipse with the given characteristics. vertices: $( 4 , - 6 ) , ( 4,10 ) \quad$ minor axis of length: 4

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

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 parabola with the given characteristic and vertex at the origin. directrix: $x = 5$