Question 1

Use the Quadratic Formula to solve for y in the following equation. $49 x ^ { 2 } - 42 x y + 9 y ^ { 2 } - 90 x - 10 y = 0$

Accepted Answer

A)  $\frac { - 21 x - 5 \pm \sqrt { - 42 x + 835 } } { 9 }$ 
B)  $\frac { 2 ( 21 x + 5 ) \pm \sqrt { - 42 x + 3340 } } { 9 }$ 
C)  $\frac { 21 x + 5 \pm \sqrt { 1020 x + 25 } } { 9 }$ 
D)  $\frac { 42 x + 10 \pm \sqrt { - 21 x + 1645 } } { 18 }$ 
E)  $\frac { - 42 x - 10 \pm \sqrt { - 21 x + 910 } } { 18 }$ 
A)  $\frac { - 21 x - 5 \pm \sqrt { - 42 x + 835 } } { 9 }$ 
B)  $\frac { 2 ( 21 x + 5 ) \pm \sqrt { - 42 x + 3340 } } { 9 }$ 
C)  $\frac { 21 x + 5 \pm \sqrt { 1020 x + 25 } } { 9 }$ 
D)  $\frac { 42 x + 10 \pm \sqrt { - 21 x + 1645 } } { 18 }$ 
E)  $\frac { - 42 x - 10 \pm \sqrt { - 21 x + 910 } } { 18 }$

Question 2

Use a graphing utility to graph the rotated conic.
$$r = \frac { 1 } { 3 - 3 \sin ( \theta - 2 \pi / 3 ) }$$
Use either grid below for your graph, whichever is more convenient.

Accepted Answer

The answer of Use a graphing utility to graph the...

Question 3

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
$$x = \left( v _ { 0 } \cos \theta \right) t \text { and } y = \left( v _ { 0 } \sin \theta \right) t - 16 t ^ { 2 } \text {. }$$
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.
$$\theta = 50 ^ { \circ } , \quad v _ { 0 } = 104 \text { feet per second }$$

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 4

Solve the following system of quadratic equations algebraically by the method of substitution.
$$\left\{ \begin{array} { l } 
- 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \
- 4 x - 2 y = 0
\end{array} \right.$$

Accepted Answer

A)  $( \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
B)  $( - \sqrt { 5 } , 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
C)  $( - \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
D)  $( - \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 3 } , 2 \sqrt { 3 } )$ 
E)  $( - \sqrt { 3 } , 2 \sqrt { 3 } ) , ( \sqrt { 3 } , - 2 \sqrt { 3 } )$ 
A)  $( \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
B)  $( - \sqrt { 5 } , 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
C)  $( - \sqrt { 5 } , - 2 \sqrt { 5 } ) , ( \sqrt { 5 } , - 2 \sqrt { 5 } )$ 
D)  $( - \sqrt { 3 } , - 2 \sqrt { 3 } ) , ( \sqrt { 3 } , 2 \sqrt { 3 } )$ 
E)  $( - \sqrt { 3 } , 2 \sqrt { 3 } ) , ( \sqrt { 3 } , - 2 \sqrt { 3 } )$

Question 5

Find the $y$-intercepts of the graph of the circle below.
$$( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$$

Accepted Answer

A)  $3 + 4 \sqrt { 2 } , 3 - 4 \sqrt { 2 }$ 
B)  $3 + 2 \sqrt { 10 } , 3 - 2 \sqrt { 10 }$ 
C)  $2 + 3 \sqrt { 5 } , 2 - 3 \sqrt { 5 }$ 
D)  $- 2 + 3 \sqrt { 3 } , - 2 - 3 \sqrt { 3 }$ 
E)  $- 2 + 3 \sqrt { 5 } , - 2 - 3 \sqrt { 5 }$ 
A)  $3 + 4 \sqrt { 2 } , 3 - 4 \sqrt { 2 }$ 
B)  $3 + 2 \sqrt { 10 } , 3 - 2 \sqrt { 10 }$ 
C)  $2 + 3 \sqrt { 5 } , 2 - 3 \sqrt { 5 }$ 
D)  $- 2 + 3 \sqrt { 3 } , - 2 - 3 \sqrt { 3 }$ 
E)  $- 2 + 3 \sqrt { 5 } , - 2 - 3 \sqrt { 5 }$

Question 6

Use the discriminant to classify the graph; then use the quadratic formula to solve for y. $$3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 0$$

Accepted Answer

A)  ellipse; $y = 3 + \sqrt { 3 } x \pm \sqrt { 4 \sqrt { 3 } x + 33 }$ 
B)  ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
C)  parabola; $y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
D)  parabola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
E)  hyperbola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
A)  ellipse; $y = 3 + \sqrt { 3 } x \pm \sqrt { 4 \sqrt { 3 } x + 33 }$ 
B)  ellipse; $\quad y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
C)  parabola; $y = 3 + \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
D)  parabola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$ 
E)  hyperbola; $y = \sqrt { 3 } x \pm \sqrt { - 4 \sqrt { 3 } x + 33 }$

Question 7

Find the graph of the following polar equation.
$$r = 3 + 2 \cos ( \theta )$$

Accepted Answer

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

Question 8

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

Accepted Answer

A)  $r = \frac { 10 } { 1 - 5 \cos \theta }$ 
B)  $r = \frac { 10 } { 1 + 5 \cos \theta }$ 
C)  $r = \frac { 2 } { 1 + 5 \cos \theta }$ 
D)  $r = \frac { 10 } { 1 + 5 \sin \theta }$ 
E)  $r = \frac { 10 } { 1 - 5 \sin \theta }$ 
A)  $r = \frac { 10 } { 1 - 5 \cos \theta }$ 
B)  $r = \frac { 10 } { 1 + 5 \cos \theta }$ 
C)  $r = \frac { 2 } { 1 + 5 \cos \theta }$ 
D)  $r = \frac { 10 } { 1 + 5 \sin \theta }$ 
E)  $r = \frac { 10 } { 1 - 5 \sin \theta }$

Question 9

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 10

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $$36 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 0$$

Accepted Answer

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

Question 11

Which answer is a polar form of the given rectangular equation?
$$16 x y = 144$$

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 12

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 13

Find the standard form of the parabola with the given characteristics.
directrix: $x = - 1$
vertex: $( 7 , - 1 )$

Accepted Answer

A)  $( x + 7 ) ^ { 2 } = 32 ( y - 1 )$ 
B)  $( x - 7 ) ^ { 2 } = 32 ( y + 1 )$ 
C)  $\quad ( x - 1 ) ^ { 2 } = 32 ( y + 7 )$ 
D)  $( y - 1 ) ^ { 2 } = 32 ( x + 7 )$ 
E)  $( y + 1 ) ^ { 2 } = 32 ( x - 7 )$ 
A)  $( x + 7 ) ^ { 2 } = 32 ( y - 1 )$ 
B)  $( x - 7 ) ^ { 2 } = 32 ( y + 1 )$ 
C)  $\quad ( x - 1 ) ^ { 2 } = 32 ( y + 7 )$ 
D)  $( y - 1 ) ^ { 2 } = 32 ( x + 7 )$ 
E)  $( y + 1 ) ^ { 2 } = 32 ( x - 7 )$

Question 14

Match the graph with its equation.

Accepted Answer

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

Question 15

Identify the center and radius of the circle below.
$$( x + 1 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$$

Accepted Answer

A)  Center: $( - 1,8 )$
Radius: 3 
B)  Center: $( - 1,8 )$
Radius: 9 
C)  Center: $( 1 , - 8 )$
Radius: 3 
D)  Center: $( 1 , - 8 )$
Radius: 9 
E)  Center: $( 8 , - 1 )$
Radius: 3 
A)  Center: $( - 1,8 )$
Radius: 3 
B)  Center: $( - 1,8 )$
Radius: 9 
C)  Center: $( 1 , - 8 )$
Radius: 3 
D)  Center: $( 1 , - 8 )$
Radius: 9 
E)  Center: $( 8 , - 1 )$
Radius: 3

Question 16

Find the standard form of the equation of the ellipse with the following characteristics.
foci: $( \pm 8,0 ) \quad$ major axis of length: 22

Accepted Answer

A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 57 } = 1$ 
B)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 121 } = 1$ 
D)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 64 } = 1$ 
E)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 420 } = 1$ 
A)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 57 } = 1$ 
B)  $\frac { x ^ { 2 } } { 121 } + \frac { y ^ { 2 } } { 64 } = 1$ 
C)  $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 121 } = 1$ 
D)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 64 } = 1$ 
E)  $\frac { x ^ { 2 } } { 484 } + \frac { y ^ { 2 } } { 420 } = 1$

Question 17

Which answer is a polar form of the given rectangular equation?
$$9 x y = 144$$

Accepted Answer

A)  $r ^ { 2 } = 16 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 4 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 4 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 16 \sin \theta \cos \theta$ 
E)  $\theta = 16 \sin \theta \cos \theta$ 
A)  $r ^ { 2 } = 16 \sec \theta \csc \theta$ 
B)  $r ^ { 2 } = 4 \sec \theta \csc \theta$ 
C)  $r ^ { 2 } = 4 \sin \theta \cos \theta$ 
D)  $r ^ { 2 } = 16 \sin \theta \cos \theta$ 
E)  $\theta = 16 \sin \theta \cos \theta$

Question 18

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola.
$2 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 0$

Accepted Answer

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

Question 19

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 20

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)  $( 3 , - 4.712 )$

Use the Quadratic Formula to solve for y in the following equation. $49 x ^ { 2 } - 42 x y + 9 y ^ { 2 } - 90 x - 10 y = 0$

Solve the following system of quadratic equations algebraically by the method of substitution. $\left\{ \begin{array} { l } - 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \\- 4 x - 2 y = 0\end{array} \right.$

Find the $y$ -intercepts of the graph of the circle below. $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$

Use the discriminant to classify the graph; then use the quadratic formula to solve for y. $3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 0$

Find the graph of the following polar equation. $r = 3 + 2 \cos ( \theta )$

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Hyperbola e=5 x=2

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $36 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 0$

Which answer is a polar form of the given rectangular equation? $16 x y = 144$

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 $( 1,0 )$ .

Find the standard form of the parabola with the given characteristics. directrix: $x = - 1$ vertex: $( 7 , - 1 )$

Match the graph with its equation.

Identify the center and radius of the circle below. $( x + 1 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$

Find the standard form of the equation of the ellipse with the following characteristics. foci: $( \pm 8,0 ) \quad$ major axis of length: 22

Which answer is a polar form of the given rectangular equation? $9 x y = 144$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $2 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 0$

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

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

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Exam 9: Topics in Analytic Geometry

Use the Quadratic Formula to solve for y in the following equation. 49x2−42xy+9y2−90x−10y=049 x ^ { 2 } - 42 x y + 9 y ^ { 2 } - 90 x - 10 y = 049x2−42xy+9y2−90x−10y=0

Use a graphing utility to graph the rotated conic. r=13−3sin⁡(θ−2π/3)r = \frac { 1 } { 3 - 3 \sin ( \theta - 2 \pi / 3 ) }r=3−3sin(θ−2π/3)1​ Use either grid below for your graph, whichever is more convenient.

Solve the following system of quadratic equations algebraically by the method of substitution. {−4x2−3y2+48=0−4x−2y=0\left\{ \begin{array} { l } - 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \\- 4 x - 2 y = 0\end{array} \right.{−4x2−3y2+48=0−4x−2y=0​

Find the yyy -intercepts of the graph of the circle below. (x−3)2+(y+2)2=36( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36(x−3)2+(y+2)2=36

Use the discriminant to classify the graph; then use the quadratic formula to solve for y. 3x2−23xy+y2+103x−6y−24=03 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 03x2−23​xy+y2+103​x−6y−24=0

Find the graph of the following polar equation. r=3+2cos⁡(θ)r = 3 + 2 \cos ( \theta )r=3+2cos(θ)

Find a polar equation of the conic with the given characteristics and with one focus at the pole. Conic Eccentricity Directrix Hyperbola e=5 x=2

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 36x2+16y2+144x+20y−335=036 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 036x2+16y2+144x+20y−335=0

Which answer is a polar form of the given rectangular equation? 16xy=14416 x y = 14416xy=144

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) .

Find the standard form of the parabola with the given characteristics. directrix: x=−1x = - 1x=−1 vertex: (7,−1)( 7 , - 1 )(7,−1)

Match the graph with its equation.

Identify the center and radius of the circle below. (x+1)2+(y−8)2=9( x + 1 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9(x+1)2+(y−8)2=9

Find the standard form of the equation of the ellipse with the following characteristics. foci: (±8,0)( \pm 8,0 ) \quad(±8,0) major axis of length: 22

Which answer is a polar form of the given rectangular equation? 9xy=1449 x y = 1449xy=144

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. 2x2+5y2+8x−2y=02 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 02x2+5y2+8x−2y=0

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

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​)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

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Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

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Use the Quadratic Formula to solve for y in the following equation. $49 x ^ { 2 } - 42 x y + 9 y ^ { 2 } - 90 x - 10 y = 0$

Solve the following system of quadratic equations algebraically by the method of substitution. $\left\{ \begin{array} { l } - 4 x ^ { 2 } - 3 y ^ { 2 } + 48 = 0 \\- 4 x - 2 y = 0\end{array} \right.$

Find the $y$ -intercepts of the graph of the circle below. $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 36$

Use the discriminant to classify the graph; then use the quadratic formula to solve for y. $3 x ^ { 2 } - 2 \sqrt { 3 } x y + y ^ { 2 } + 10 \sqrt { 3 } x - 6 y - 24 = 0$

Find the graph of the following polar equation. $r = 3 + 2 \cos ( \theta )$

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

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $36 x ^ { 2 } + 16 y ^ { 2 } + 144 x + 20 y - 335 = 0$

Which answer is a polar form of the given rectangular equation? $16 x y = 144$

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 $( 1,0 )$ .

Find the standard form of the parabola with the given characteristics. directrix: $x = - 1$ vertex: $( 7 , - 1 )$

Identify the center and radius of the circle below. $( x + 1 ) ^ { 2 } + ( y - 8 ) ^ { 2 } = 9$

Find the standard form of the equation of the ellipse with the following characteristics. foci: $( \pm 8,0 ) \quad$ major axis of length: 22

Which answer is a polar form of the given rectangular equation? $9 x y = 144$

Classify the graph of the equation below as a circle, a parabola, an ellipse, or a hyperbola. $2 x ^ { 2 } + 5 y ^ { 2 } + 8 x - 2 y = 0$

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