Question 1

Identify the type of graph.
-$6 x ^ { 2 } - 12 x + 3 y ^ { 2 } + 18 y - 21 = 0$

Accepted Answer

A)  The graph is the point (-1, 3). 
B)  The graph is a circle with center (1, -3). 
C)  The equation has no graph. 
D)  The graph is an ellipse with center (1, -3). 
A)  The graph is the point (-1, 3). 
B)  The graph is a circle with center (1, -3). 
C)  The equation has no graph. 
D)  The graph is an ellipse with center (1, -3).

Question 2

Write an equation for the hyperbola.
-vertices $( - 6,2 ) , ( - 6,14 ) ; e = \frac { 2 \sqrt { 13 } } { 6 }$

Accepted Answer

A)  $\frac { ( y - 6 ) ^ { 2 } } { 36 } - \frac { ( x + 8 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( \mathrm { y } - 6 ) ^ { 2 } } { 36 } - \frac { ( \mathrm { x } - 8 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( y + 6 ) ^ { 2 } } { 16 } - \frac { ( x - 8 ) ^ { 2 } } { 36 } = 1$ 
D)  $\frac { ( y - 8 ) ^ { 2 } } { 36 } - \frac { ( x + 6 ) ^ { 2 } } { 16 } = 1$ 
A)  $\frac { ( y - 6 ) ^ { 2 } } { 36 } - \frac { ( x + 8 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( \mathrm { y } - 6 ) ^ { 2 } } { 36 } - \frac { ( \mathrm { x } - 8 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( y + 6 ) ^ { 2 } } { 16 } - \frac { ( x - 8 ) ^ { 2 } } { 36 } = 1$ 
D)  $\frac { ( y - 8 ) ^ { 2 } } { 36 } - \frac { ( x + 6 ) ^ { 2 } } { 16 } = 1$

Question 3

Determine the two equations necessary to graph the hyperbola with a graphing calculator.
-$$y ^ { 2 } - 64 x ^ { 2 } = 64$$

Accepted Answer

A)  $y = 8 + \sqrt { x ^ { 2 } }$ and $y = 8 - \sqrt { x ^ { 2 } }$ 
B)  $y = 8 + \sqrt { x ^ { 2 } }$ and $- y = - 8 - \sqrt { x ^ { 2 } }$ 
C)  $y = 8 x ^ { 2 } + 8$ and $y = - 8 x ^ { 2 } - 8$ 
D)  $y = 8 \sqrt { x ^ { 2 } + 1 }$ and $y = - 8 \sqrt { x ^ { 2 } + 1 }$ 
A)  $y = 8 + \sqrt { x ^ { 2 } }$ and $y = 8 - \sqrt { x ^ { 2 } }$ 
B)  $y = 8 + \sqrt { x ^ { 2 } }$ and $- y = - 8 - \sqrt { x ^ { 2 } }$ 
C)  $y = 8 x ^ { 2 } + 8$ and $y = - 8 x ^ { 2 } - 8$ 
D)  $y = 8 \sqrt { x ^ { 2 } + 1 }$ and $y = - 8 \sqrt { x ^ { 2 } + 1 }$

Question 4

Provide an appropriate response.
-It can be shown that an ellipse with foci $( \mathrm { c } , 0 )$ and $( - \mathrm { c } , 0 )$ where the sum of the distances from any point $( x , y )$ of the ellipse to the two foci is $2 a$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { a ^ { 2 } - c ^ { 2 } } = 1$. Use this result to find an equation of an ellipse with foci $( 3,0 )$ and $( - 3,0 )$, where the sum of the distances from any point of the ellipse to the two foci is 8 .

Accepted Answer

A)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 13 } = 1$ 
B)  $x ^ { 2 } + y ^ { 2 } = 64$ 
C)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$ 
A)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 13 } = 1$ 
B)  $x ^ { 2 } + y ^ { 2 } = 64$ 
C)  $\frac { x ^ { 2 } } { - 3 } + \frac { y ^ { 2 } } { 64 } = 1$ 
D)  $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$

Question 5

Provide an appropriate response.
-Fill in the blanks to complete the statement. The graph of $\mathrm { P } ( \mathrm { x } ) = - 0.8 ( \mathrm { x } - 2 ) ^ { 2 } - 3$ is a parabola opening......... and is ........than that of $y = x ^ { 2 }$.

Accepted Answer

A)  up; narrower 
B)  down; wider 
C)  down; narrower 
D)  up; wider 
A)  up; narrower 
B)  down; wider 
C)  down; narrower 
D)  up; wider

Question 6

Graph the ellipse.
-$$\frac { ( x + 2 ) ^ { 2 } } { 25 } + \frac { ( y - 4 ) ^ { 2 } } { 9 } = 1$$

Accepted Answer

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

Question 7

Identify the type of graph.
-$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 36 } = 1$

Accepted Answer

A)  The graph is a circle with center at (6, 6) . 
B)  The graph is an ellipse with foci on the x-axis . 
C)  The graph is a circle with center at (36, 36) . 
D)  The graph is a circle with center at (0, 0) . 
A)  The graph is a circle with center at (6, 6) . 
B)  The graph is an ellipse with foci on the x-axis . 
C)  The graph is a circle with center at (36, 36) . 
D)  The graph is a circle with center at (0, 0) .

Question 8

Match the equation of the parabola with the appropriate description.
-$$y - 2 = 2 ( x - 6 ) ^ { 2 }$$

Accepted Answer

A)  Vertex at $( 12,2 )$ 
B)  Vertex at $( 6,2 )$ 
C)  Vertex at $( - 12 , - 2 )$ 
D)  Vertex at $( - 6 , - 2 )$ 
A)  Vertex at $( 12,2 )$ 
B)  Vertex at $( 6,2 )$ 
C)  Vertex at $( - 12 , - 2 )$ 
D)  Vertex at $( - 6 , - 2 )$

Question 9

Give the focus, directrix, and axis for the parabola.
-$$y ^ { 2 } = - 20 x$$

Accepted Answer

A)  $( 5,0 ) , x = - 5$, $x$-axis 
B)  $( 0 , - 5 ) , y = 5 , y$-axis 
C)  $( - 5,0 ) , x = 5$, $x$-axis 
D)  $( - 5,0 ) , y = 5 , y$-axis 
A)  $( 5,0 ) , x = - 5$, $x$-axis 
B)  $( 0 , - 5 ) , y = 5 , y$-axis 
C)  $( - 5,0 ) , x = 5$, $x$-axis 
D)  $( - 5,0 ) , y = 5 , y$-axis

Question 10

Write an equation for the ellipse.
-foci at $( 1,7 ) , ( 1,1 )$; major axis length of 10

Accepted Answer

A)  $\frac { ( y - 4 ) ^ { 2 } } { 25 } + \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( y - 4 ) ^ { 2 } } { 25 } + \frac { ( x + 1 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y + 1 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { 25 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$ 
A)  $\frac { ( y - 4 ) ^ { 2 } } { 25 } + \frac { ( x - 1 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( y - 4 ) ^ { 2 } } { 25 } + \frac { ( x + 1 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 4 ) ^ { 2 } } { 16 } + \frac { ( y + 1 ) ^ { 2 } } { 25 } = 1$ 
D)  $\frac { ( x - 4 ) ^ { 2 } } { 25 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$

Question 11

Write an equation for the ellipse.
-minor axis from $( - 3,3 )$ to $( 1,3 )$; major axis from $( - 1 , - 1 )$ to $( - 1,7 )$

Accepted Answer

A)  $\frac { ( \mathrm { x } - 2 ) ^ { 2 } } { 4 } + \frac { ( \mathrm { y } - 4 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 3 ) ^ { 2 } } { 4 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$ 
D)  $\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$ 
A)  $\frac { ( \mathrm { x } - 2 ) ^ { 2 } } { 4 } + \frac { ( \mathrm { y } - 4 ) ^ { 2 } } { 16 } = 1$ 
B)  $\frac { ( x + 1 ) ^ { 2 } } { 4 } + \frac { ( y - 3 ) ^ { 2 } } { 16 } = 1$ 
C)  $\frac { ( x - 3 ) ^ { 2 } } { 4 } + \frac { ( y + 1 ) ^ { 2 } } { 16 } = 1$ 
D)  $\frac { ( x - 1 ) ^ { 2 } } { 4 } + \frac { ( y + 3 ) ^ { 2 } } { 16 } = 1$

Question 12

Choose the equation that matches the graph.
-

Accepted Answer

A)  $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 36 } = 1$ 
D)  $\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 36 } = 1$ 
A)  $\frac { y ^ { 2 } } { 36 } - \frac { x ^ { 2 } } { 4 } = 1$ 
B)  $\frac { x ^ { 2 } } { 36 } - \frac { y ^ { 2 } } { 4 } = 1$ 
C)  $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 36 } = 1$ 
D)  $\frac { y ^ { 2 } } { 4 } - \frac { x ^ { 2 } } { 36 } = 1$

Question 13

Graph the parabola.
-$$x = \frac { 5 } { 7 } ( y - 4 ) ^ { 2 } + 2$$

Accepted Answer

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

Question 14

Provide an appropriate response.
-It can be shown that a hyperbola with center at the origin, foci at $F ^ { \prime } ( - c , 0 )$ and $F ( c , 0 )$, and equation $d \left( P , F ^ { \prime } \right) - d ( P , F ) = 2 a$ has equation $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. Use this result to find an equation of a hyperbola with center at the origin, foci at $( - 8,0 )$ and $( 8,0 )$, and absolute value of the distances from any point of the hyperbola to the two foci equal to 8 .

Accepted Answer

A)  $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 80 } = 1$ 
B)  $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 8 } = 1$ 
C)  $x ^ { 2 } - y ^ { 2 } = 8$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 1$ 
A)  $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 80 } = 1$ 
B)  $\frac { x ^ { 2 } } { 64 } - \frac { y ^ { 2 } } { 8 } = 1$ 
C)  $x ^ { 2 } - y ^ { 2 } = 8$ 
D)  $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 48 } = 1$

Question 15

Write an equation for the hyperbola.
-vertices at $( 0 , \pm 10 )$, asymptotes at $y = \pm \frac { 5 } { 4 } x$

Accepted Answer

A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 25 } = 1$ 
B)  $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 64 } = 1$ 
C)  $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 16 } = 1$ 
D)  $\frac { y ^ { 2 } } { 64 } - \frac { x ^ { 2 } } { 100 } = 1$ 
A)  $\frac { y ^ { 2 } } { 16 } - \frac { x ^ { 2 } } { 25 } = 1$ 
B)  $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 64 } = 1$ 
C)  $\frac { y ^ { 2 } } { 100 } - \frac { x ^ { 2 } } { 16 } = 1$ 
D)  $\frac { y ^ { 2 } } { 64 } - \frac { x ^ { 2 } } { 100 } = 1$

Question 16

Provide an appropriate response.
-You have to draw an ellipse such that the length of the major axis is fixed at 20 cm. The 
length of the minor axis can have the values 10, 14, or 16 cm. You are told that your ellipse
should be as flat as possible. Which value of length of minor axis would you choose and
why? (Do not calculate).

Accepted Answer

The eccentricity e @#LAT-DLM& should be

Question 17

Identify the equation as a parabola, circle, ellipse, or hyperbola.
-$4 x ^ { 2 } = 9 y ^ { 2 } + 36$

Accepted Answer

A)  Circle 
B)  Parabola 
C)  Hyperbola 
D)  Ellipse 
A)  Circle 
B)  Parabola 
C)  Hyperbola 
D)  Ellipse

Question 18

Identify the type of graph.
-$$- x ^ { 2 } - 10 x + 6 y ^ { 2 } + 12 y = 16$$

Accepted Answer

A)  The graph is an ellipse with center $\left( - \frac { 6 } { 5 } , 2 \right)$. 
B)  The graph is a circle with center $\left( \frac { 6 } { 5 } , - 2 \right)$. 
C)  The equation has no graph. 
D)  The graph is a hyperbola opening to the right and left. 
A)  The graph is an ellipse with center $\left( - \frac { 6 } { 5 } , 2 \right)$. 
B)  The graph is a circle with center $\left( \frac { 6 } { 5 } , - 2 \right)$. 
C)  The equation has no graph. 
D)  The graph is a hyperbola opening to the right and left.

Question 19

Identify the type of conic section.
-Identify the type of conic section consisting of the set of all points in the plane for which the absolute value of the difference of the distances from the points (0, 7) and (0, -7) is 3.

Accepted Answer

A)  Parabola 
B)  Hyperbola 
C)  Circle 
D)  Ellipse 
A)  Parabola 
B)  Hyperbola 
C)  Circle 
D)  Ellipse

Question 20

Graph the equation. Give the domain and range. Identify whether the equation is a function.
-$\frac{y}{6}=\sqrt{1+\frac{x^{2}}{16}}$

Accepted Answer

A) 
 
$( \infty , \infty ) ; ( \infty , - 6 ] ;$ not a function 
B) 
 
$( \infty , \infty )$; $[ 1 , \infty )$; function 
C) 
 
$( \infty , \infty ) ; ( \infty , - 1 ]$; not a function 
D)

$( \infty , \infty ) ; [ 6 , \infty )$; function 
A) 
 
$( \infty , \infty ) ; ( \infty , - 6 ] ;$ not a function 
B) 
 
$( \infty , \infty )$; $[ 1 , \infty )$; function 
C) 
 
$( \infty , \infty ) ; ( \infty , - 1 ]$; not a function 
D)

$( \infty , \infty ) ; [ 6 , \infty )$; function

Identify the type of graph. - $6 x ^ { 2 } - 12 x + 3 y ^ { 2 } + 18 y - 21 = 0$

Write an equation for the hyperbola. -vertices $( - 6,2 ) , ( - 6,14 ) ; e = \frac { 2 \sqrt { 13 } } { 6 }$

Determine the two equations necessary to graph the hyperbola with a graphing calculator. - $y ^ { 2 } - 64 x ^ { 2 } = 64$

Provide an appropriate response. -Fill in the blanks to complete the statement. The graph of $\mathrm { P } ( \mathrm { x } ) = - 0.8 ( \mathrm { x } - 2 ) ^ { 2 } - 3$ is a parabola opening......... and is ........than that of $y = x ^ { 2 }$ .

Graph the ellipse. - $\frac { ( x + 2 ) ^ { 2 } } { 25 } + \frac { ( y - 4 ) ^ { 2 } } { 9 } = 1$ $Graph the ellipse. - \frac { ( x + 2 ) ^ { 2 } } { 25 } + \frac { ( y - 4 ) ^ { 2 } } { 9 } = 1$

Identify the type of graph. - $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 36 } = 1$

Match the equation of the parabola with the appropriate description. - $y - 2 = 2 ( x - 6 ) ^ { 2 }$

Give the focus, directrix, and axis for the parabola. - $y ^ { 2 } = - 20 x$

Write an equation for the ellipse. -foci at $( 1,7 ) , ( 1,1 )$ ; major axis length of 10

Write an equation for the ellipse. -minor axis from $( - 3,3 )$ to $( 1,3 )$ ; major axis from $( - 1 , - 1 )$ to $( - 1,7 )$

Choose the equation that matches the graph. -

Graph the parabola. - $x = \frac { 5 } { 7 } ( y - 4 ) ^ { 2 } + 2$ $Graph the parabola. - x = \frac { 5 } { 7 } ( y - 4 ) ^ { 2 } + 2$

Write an equation for the hyperbola. -vertices at $( 0 , \pm 10 )$ , asymptotes at $y = \pm \frac { 5 } { 4 } x$

Identify the equation as a parabola, circle, ellipse, or hyperbola. - $4 x ^ { 2 } = 9 y ^ { 2 } + 36$

Identify the type of graph. - $- x ^ { 2 } - 10 x + 6 y ^ { 2 } + 12 y = 16$

Identify the type of conic section. -Identify the type of conic section consisting of the set of all points in the plane for which the absolute value of the difference of the distances from the points (0, 7) and (0, -7) is 3.

Graph the equation. Give the domain and range. Identify whether the equation is a function. - $\frac{y}{6}=\sqrt{1+\frac{x^{2}}{16}}$ $Graph the equation. Give the domain and range. Identify whether the equation is a function. - \frac{y}{6}=\sqrt{1+\frac{x^{2}}{16}}$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Further Topics in Algebra

Filters

Exam 11: Analytic Geometry

Identify the type of graph. - 6x2−12x+3y2+18y−21=06 x ^ { 2 } - 12 x + 3 y ^ { 2 } + 18 y - 21 = 06x2−12x+3y2+18y−21=0

Write an equation for the hyperbola. -vertices (−6,2),(−6,14);e=2136( - 6,2 ) , ( - 6,14 ) ; e = \frac { 2 \sqrt { 13 } } { 6 }(−6,2),(−6,14);e=6213​​

Determine the two equations necessary to graph the hyperbola with a graphing calculator. - y2−64x2=64y ^ { 2 } - 64 x ^ { 2 } = 64y2−64x2=64

Provide an appropriate response. -Fill in the blanks to complete the statement. The graph of P(x)=−0.8(x−2)2−3\mathrm { P } ( \mathrm { x } ) = - 0.8 ( \mathrm { x } - 2 ) ^ { 2 } - 3P(x)=−0.8(x−2)2−3 is a parabola opening......... and is ........than that of y=x2y = x ^ { 2 }y=x2 .

Graph the ellipse. - (x+2)225+(y−4)29=1\frac { ( x + 2 ) ^ { 2 } } { 25 } + \frac { ( y - 4 ) ^ { 2 } } { 9 } = 125(x+2)2​+9(y−4)2​=1

Identify the type of graph. - x236+y236=1\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 36 } = 136x2​+36y2​=1

Match the equation of the parabola with the appropriate description. - y−2=2(x−6)2y - 2 = 2 ( x - 6 ) ^ { 2 }y−2=2(x−6)2

Give the focus, directrix, and axis for the parabola. - y2=−20xy ^ { 2 } = - 20 xy2=−20x

Write an equation for the ellipse. -foci at (1,7),(1,1)( 1,7 ) , ( 1,1 )(1,7),(1,1) ; major axis length of 10

Write an equation for the ellipse. -minor axis from (−3,3)( - 3,3 )(−3,3) to (1,3)( 1,3 )(1,3) ; major axis from (−1,−1)( - 1 , - 1 )(−1,−1) to (−1,7)( - 1,7 )(−1,7)