Slide 1 / 160 Pre-Calc Conics www.njctl.org 2015-03-24 Slide 2 / 160 Table of Contents Review of Midpoint and Distance Formulas Parabolas Circles Ellipses Hyperbolas Recognizing Conic Sections from the General Form Intro to Conic Sections click on the topic to go to that section Slide 3 / 160
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Note: To convert into Standard Form, we use a process called Completing the Square.
Steps:
1) Group the quadratic and its linear term on one side, and move the other linear and constant terms to the other side.
2) If there is a number in front of the quadratic, factor it out of the group.
3) Take the number in front of the linear term, divide it in half and square it.
4) Add this number inside the parenthesis; multiply it by the number you factored out in step two, and add it to the other side of the equation as well.
5) Factor the quadratic function inside the parenthesis
Parabolas
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Parabolas
Example: Find the vertex of the parabola
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17
A (4, 5)
B (-4, 5)
C (-5, 4)
D (5, 4)
What is the vertex of y2 - 10y - x + 29 = 0?
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Parabolas
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18 What is the vertex of
A (4, 5)
B (-4, 5)
C (-5, 4)
D (5, 4)
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Parabolas
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Converting from General Form to Standard Form
}
+18
}-12
Parabolas
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19 What should be factored out of (4y2 - 8y + ___) = x - 9 + ___ ? Te
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rTe
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Parabolas
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20 What value completes the square of 4(y2 - 2y + ___) = x - 9 + ___ ? Te
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rTe
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r
Parabolas
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21 What value should follow "-9" in 4(y2 - 2y + ___) = x - 9 + ___ ? Te
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rTe
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r
Parabolas
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22 Which is the correct standard form of4(y2 - 2y + ___) = x - 9 + ___
A
B
C
D
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Parabolas
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23 What should be factored out of (-5x2 - 20x + ___) = y - 7 + ___ ? Te
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Parabolas
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24 What value completes the square of -5(x2 + 4x + ___) = y - 7 + ___ ?
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25 What value should follow "-7" in -5(x2 + 4x + ___) = y - 7 + ___?
Teac
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Parabolas
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26 Which is the correct standard form of(-5x2 - 20x + ___) = y - 7 + ___
A
B
C
D
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Parabolas
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Focus and Directrix of a Parabola
Axis of Symmetry
Directrix
Focus
Every point on the parabola is the same distance from the directrix and the focus.
L1
L2
L1=L2
The focal distance is the distance from the vertex to the focus, which is the same as the distance from the vertex to the directrix.
Parabolas
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Directrix
Focus
L1
L2
L1=L2
Eccentricity of a Parabola
Parabolas
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Parts of a ParabolaWhether a quadratic has the x2 or y2, they have the same parts.
Directrix
Axis of Symmetry
Vertex
FocusVertex
Focus
Directrix
Axis of Symmetry
cy2+dy+bx+e=0ax2+bx+dy+e=0
Parabolas
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Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?
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Parabolas
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Graph the equation from the last example.Parabolas
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Identify the vertex and the focus, the equations for the axis of symmetry and the directrix, and the direction of the opening of the parabola with the given equation. What is the parabola's eccentricity?
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Parabolas
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Graph Teac
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Parabolas
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GraphTe
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Parabolas
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27 Given the following equation, which direction does it open?
A UP
B DOWN
C LEFT
D RIGHT
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Parabolas
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28 Where is the vertex for the following equation?
A (-3 , 4)
B (3 , 4)
C (4 , 3)
D (4 , -3)
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Parabolas
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29 What is the equation of the axis of symmetry for the following equation?
A y = 3
B y = -3
C x = 4
D x = -4
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Parabolas
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30 What is the focal distance in the following equation?
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Parabolas
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31 What is the equation of the directrix for the following equation?
A y = 2
B y = -4
C x = 3
D x = -5
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Parabolas
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32 Where is the focus for the following equation?
A (-3 , 5)
B (3 , 5)
C (5 , 3)
D (5 , -3)
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Parabolas
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33 What is the eccentricity of the following conic section?
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Parabolas
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42 Where is the vertex for the following equation?
A (0 , 4)
B (0 , -4)
C (4 , 0)
D (-4 , 0)
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Parabolas
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43 What is the equation of the axis of symmetry for the following equation?
A y = 0
B y = -0
C x = 4
D x = -4
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Parabolas
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44 What is the focal distance in the following equation?Te
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Parabolas
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45 What is the equation of the directrix for the following equation?
A y = 0
B y = -4
C x = 8
D x = 0
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Parabolas
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46 Where is the focus for the following equation?
A (4 , 8)
B (-4 , 4)
C (4 , 4)
D (4 , -4)
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Parabolas
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47 What is the eccentricity of the following conic section?Te
An ellipse is the set of points the same total distance from 2 points.
In this example,
As point moves along theellipse, L1 and L2 will changebut their sum will stay ten.
P
Ellipses
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P
In this graph F1 and F2 are foci. (Plural of focus)
They lie on the major axis. (The longest distance)
The shortest distance is the minor axis.
Where the axes intersect is the ellipse'scenter.
The more elongated the ellipse the closerthe eccentricity is to 1. The closer an ellipse is to being a circle, the closer theeccentricity is to 0. (0 < e < 1)
Ellipses
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AB
CD E
60 What letter or letters corresponds with ellipse's center?
A
B
C
D
E
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Ellipses
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AB
CD E
61 What letter or letters corresponds with ellipse's foci?
A
B
C
D
E
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Ellipses
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AB
CD E
62 What letter or letters corresponds with ellipse's major axis?
A
B
C
D
E
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Ellipses
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63 Which choice best describes an ellipse's eccentricity?
A e = 0
B 0< e < 1
C e = 1
D e > 1
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Ellipses
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64 What is the center of
A (9 , 4)
B (5 , 6)
C (-5 , -6)
D (3 , 2)
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Ellipses
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65 How long is the major axis of
A 9
B 4
C 3
D 2
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Ellipses
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66 How long is the minor axis of
A 9
B 4
C 3
D 2
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Ellipses
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67 Name one foci of
A
B
C
D
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Ellipses
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68 Name one foci of
A
B
C
D
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Ellipses
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Graphing an Ellipse
· Find and graph the center · Find the length and direction of the major and minor axes· From the center go half the length the axis from the center for each· Graph the ellipse
The center is (4 , -2)The major axis is 6 units and horizontalThe minor axis is 4 units and vertical
Ellipses
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What is equation of an ellipse with foci (3, -2) and (3 , 6) and minor axis of length 8?
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Ellipses
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69 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, what is the center of the ellipse?
A (8 , 2)
B (0 , 2)
C (0 , 1)
D (-8 , 1)
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Ellipses
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70 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, in which direction is the ellipse elongated?
A horizontally
B vertically
C obliquely
D it is not elongated
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Ellipses
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71 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, how far is it from the center to an endpoint of the major axis?
A 10
B 100
C 5
D 25
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Ellipses
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72 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, which equation would be used to find the distance from the center to an endpoint of the minor axis?
A
B
C
D
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Ellipses
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73 Given that an ellipse has foci (4 , 1) and (-4 , 1) and major axis of length 10, find a.
A
B
C
D
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Ellipses
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Converting to Standard Form· complete the square for x and/or y· factor the x's and y's· divide by the constant
Ex:
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Ellipses
Ex:
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75 Convert the following ellipse to standard form.
To graph a hyperbola in standard form:· graph (h,k) as center of graph
· go a right and left of the center, and b up and down
· make a rectangle through the four points from previous step
· draw asymptotes that contain the diagonals of the rectangle
· decide if hyperbola goes left & right or up & down left & right: the "x term" is first up & down: the "y term" is first
· graph hyperbola
Hyperbolas
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Example: Graph
What are the slopes of the asymptotes? How does this relate to a and b? Why?
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The cente r of the rectangle is?
From the cente r move le ft/right?
From the cente r move up/down?
The hyperbola opens?
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Example: Graph
The hyperbola opens?
From the cente r move up/down?
From the cente r move le ft/right?
The cente r of the rectangle is?
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Hyperbolas
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Standard Form of an Hyperbola
The Foci are equidistant from the center in the horizontal direction if the x-term comes first, or in the vertical direction if the y-term comes first.
The distance from the center to the foci is
In this example, the focal distance is
And their location is at and
Hyperbolas
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85 What is the focal distance for the following equation?
A 12B 13C 5D 8
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Hyperbolas
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86 What is the location of one of the foci for this hyperbola?
A (-13,-6)B (10,-6)C (-10,-6)D (13,-6)
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Hyperbolas
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87 What is the location of one of the foci for this hyperbola?