Conic Sections Curves with second degree Equations

Conic Shapes  Conic shapes are obtained by “slicing a cone” or a “double cone” intersecting at the vertex  Different “slices” will obtain different curves  The 4 basic curves are : parabola, circle, ellipse, hyperbola

Conic Shapes  Parabola  This shape is obtained by “slicing a cone” by “slicing a cone” at an angle to at an angle to the “axis” of the “axis” of the cones the cones

Conic Shapes  Circle  This shape is obtained by “slicing a cone” by “slicing a cone” perpendicular to perpendicular to the axis of the the axis of the cones cones

Conic Shapes  Ellipse  This shape is obtained by “slicing a cone” by “slicing a cone” at an angle through at an angle through the axis of the cone the axis of the cone

Conic Shapes  Hyperbola  This shape is obtained by “slicing both cones” “slicing both cones” in a slice parallel to in a slice parallel to to the axis of the to the axis of the cones cones

Parabola  Definition: Set of all points that are equidistant from a given point (focus) and a given line (directrix)  The vertex is exactly ½ way between the ½ way between the focus and directrix. The parabola curves The parabola curves around the focus around the focus

Parabola  Graphing a Parabola  The simple equation is: y = 1 / (4p) x 2 or x 2 = 4py y = 1 / (4p) x 2 or x 2 = 4py “p” is the distance “p” is the distance from the vertex to from the vertex to either the focus either the focus or directrix or directrix

Parabola  The parabola with the equation of y = 1 / 8 x 2 has the following points on its graph: x y x y

Parabola  The parabola with the equation of y = 1 / 8 x 2 has a vertex at the point (0,0). In the equation p = 2. p = 2. This means the This means the focus is 2 units focus is 2 units above the vertex above the vertex or at the point (0,2). or at the point (0,2).

Parabola  The parabola with the equation of y = 1 / 8 x 2 has a vertex at the point (0,0). In the equation p = 2. p = 2. This means the This means the directrix is directrix is 2 units below 2 units below the vertex and is the the vertex and is the line with the equation line with the equation y = -2 y = -2

Parabola  If the equation has x 2, then it is a veritcal parabola.  If the equation has y 2, then it is a horizontal parabola.

Parabola  If 1 / (4p) is positive, then the parabola is going in a positive direction (up if vertical, right if horizontal).  If 1 / (4p) is negative, then the parabola is going in a negative direction (down if vertical, left if horizontal)

Parabola Example Parabola Example  For the parabola with the equation:  y = 2x 2  Find the Vertex  Find p  Find the focus  Find the directrix  Make a table showing 5 points

Parabola Example Parabola Example  For the parabola with the equation:  y = 2x 2 Points:  Vertex (0,0) x y  p = 1 / 8 because 1 / (4p) =  focus (0, 1 / 8 ) -1 2  directrix y = - 1 /

Parabola Example Parabola Example  For the parabola with the equation:  y = 2x 2  Vertex (0,0)  p = 1 / 8  focus (0, 1 / 8 )  directrix y = - 1 / 8

Circle  Definition: Set of all points equidistant from a given point (center). The distance is called the radius. r r

Circle  Graphing the circle:  The simple equation is x 2 + y 2 = r 2 is x 2 + y 2 = r 2 The center for r The center for r this circle is (0,0) r r this circle is (0,0) r r and its radius is r r and its radius is r r

Circle  Graphing the circle:  Given the equation : x 2 + y 2 = 16  Give the center  Give the radius  Give 4 points  Graph

Circle  Graphing the circle:  Given the equation : x 2 + y 2 = 16  center (0,0) (0,4)  radius 4  Give 4 points (4,0), (-4,0) (-4,0) (4,0) (4,0), (-4,0) (-4,0) (4,0) (0,4), (0,-4) (0,4), (0,-4) (0,-4) (0,-4)

Ellipse  Definition: The set of all points, so that the sum of the distances of each point from 2 given points is constant  The 2 given points are called foci are called foci

Ellipse Graphing the Ellipse The simple equation is: x 2 y 2 a 2 b 2

Ellipse Graphing the Ellipse In the equation, a is the horizontal distance the ellipse is from the center

Ellipse Graphing the Ellipse In the equation, b is the vertical distance the ellipse is from the center

Ellipse Graphing the Ellipse The foci c are on the longest axis of the ellipse. To find c, c 2 = a 2 – b 2 or c 2 = b 2 – a 2

Ellipse  The ellipse with the equation x 2 y 2 x 2 y has the center has the center at (0,0) at (0,0)

Ellipse  The ellipse with the equation x 2 y 2 x 2 y has a horizontal has a horizontal distance of 5 each distance of 5 each way from the center way from the center

Ellipse  The ellipse with the equation x 2 y 2 x 2 y has a vertical 3 has a vertical 3 distance of 3 each -3 distance of 3 each -3 way from the center way from the center

Ellipse  The ellipse with the equation x 2 y 2 x 2 y has the foci at has the foci at (-4,0) and (4,0) -4 4 (-4,0) and (4,0) -4 4 because c 2 = 25 – 9 because c 2 = 25 – 9 so c = 4 so c = 4

Ellipse  If a 2 is larger, the ellipse is a horizontal ellipse and the foci are on the horizontal axis

Ellipse  If b 2 is larger, the ellipse is a vertical ellipse and the foci are on the foci are on the vertical axis vertical axis

Ellipse  The longest axis is called the Major Axis  The shortest axis is called the Minor Axis

Ellipse Example Ellipse Example  Graphing the Ellipse  Given the equation: x 2 y 2 x 2 y Give the Center Give the Center Give the Vertices Give the Vertices Give the Co-Vertices Give the Co-Vertices Give the Foci Give the Foci

Ellipse Example Ellipse Example  Graphing the Ellipse  Given the equation: x 2 y 2 x 2 y Center (0,0) Center (0,0) Vertices (on the longest axis) (5,0) & (-5,0) Vertices (on the longest axis) (5,0) & (-5,0) Co-Vertices (on the shortest axis) Co-Vertices (on the shortest axis) (0,3) & (0,-3) (0,3) & (0,-3)

Ellipse Example Ellipse Example  Graphing the Ellipse  Given the equation: x 2 y 2 x 2 y Foci would be c where c 2 = a 2 – b 2 Foci would be c where c 2 = a 2 – b 2 c 2 = 25 – 9 = 16, so c = 4 c 2 = 25 – 9 = 16, so c = 4

Ellipse Example Ellipse Example  Graphing the Ellipse  Given the equation: (0,3) x 2 y 2 x 2 y (-5,0) (5,0) 25 9 (-5,0) (5,0) Center (0,0) (-4,0) (4,0) Center (0,0) (-4,0) (4,0) Vertices (5,0) & (-5,0) (0,-3) Vertices (5,0) & (-5,0) (0,-3) Co-Vertices (0,3) & (0,-3) Co-Vertices (0,3) & (0,-3) Foci (4,0), (-4,0) Foci (4,0), (-4,0)

Hyperbola  Definition: The set of all points so that the difference of the distances of the points from 2 given points is constant.  The 2 given points are points are called foci.

Hyperbola  Graphing the Hyperbola:  The simple equation is  x 2 y 2 a 2 b 2 a 2 b 2 or or y 2 x 2 y 2 x 2 b 2 a 2 b 2 a 2

Hyperbola  Graphing the Hyperbola:  If the equation has x 2 positive, then  x 2 y 2 a 2 b 2 a 2 b 2 and the and the hyperbola hyperbola is horizontal is horizontal

Hyperbola  Graphing the Hyperbola:  If the equation has y 2 positive, then  y 2 x 2 b 2 a 2 b 2 a 2 and the and the hyperbola hyperbola is vertical is vertical

Hyperbola  If the equation is  x 2 y 2 a 2 b 2 a 2 b 2 then the then the horizontal horizontal hyperbola hyperbola has vertices of (-a,0) and (a,0) has vertices of (-a,0) and (a,0)

Hyperbola  If the equation is  x 2 y 2 a 2 b 2 a 2 b 2 then the foci then the foci are on the are on the horizontal axis farther from the origin than the vertices horizontal axis farther from the origin than the vertices

Hyperbola  Graphing the Hyperbola:  If the equation is  y 2 x 2 b 2 a 2 b 2 a 2 then the then the vertices are vertices are (0,b) and (0,-b) (0,b) and (0,-b)

Hyperbola  To find the value of c, for the foci:  c 2 = a 2 + b 2

Hyperbola  Graphing the Hyperbola:  If the simple equation is  x 2 y 2 Then there are 2 a 2 b 2 lines that tell how a 2 b 2 lines that tell how orwide the hyperbola orwide the hyperbola y 2 x 2 curves will be. y 2 x 2 curves will be. b 2 a 2 They are called b 2 a 2 They are called asymptotes.

Hyperbola Hyperbola  Asymptotes- are lines the curve gets closer and closer to but never touches

Hyperbola  Graphing the Hyperbola:  If the simple equation is  x 2 y 2 The equation of a 2 b 2 the asymptotes for a 2 b 2 the asymptotes for oreither equation is oreither equation is y 2 x 2 y = + b / a x y 2 x 2 y = + b / a x b 2 a 2 b 2 a 2

Hyperbola  Graphing the Hyperbola:  Given the equation: x 2 y 2 x 2 y Find the vertices Find the vertices Find the foci Find the foci Give the equation of the asymptotes Give the equation of the asymptotes

Hyperbola  Graphing the Hyperbola:  Given the equation: x 2 y 2 The vertices are x 2 y 2 The vertices are 16 9 (4,0) & (-4,0) 16 9 (4,0) & (-4,0) because the x 2 term is positive and a 2 = 16, so a = 4

Hyperbola  Graphing the Hyperbola:  Given the equation: x 2 y 2 Since b 2 = 9, b = 3 x 2 y 2 Since b 2 = 9, b = The foci are found by finding c, which is c 2 = a 2 + b 2, or c 2 = ; which means c = 5, the foci are (-5,0) and (5,0) 16 9 The foci are found by finding c, which is c 2 = a 2 + b 2, or c 2 = ; which means c = 5, the foci are (-5,0) and (5,0)

Hyperbola  Graphing the Hyperbola:  Given the equation: x 2 y 2 The equation of the x 2 y 2 The equation of the 16 9 asymptotes is 16 9 asymptotes is y = + b / a x Since a = 4 and b = 3, the equation of the asymptotes is y = + 3 / 4 x of the asymptotes is y = + 3 / 4 x

Hyperbola  Graphing the Hyperbola:  Given the equation: x 2 y 2 x 2 y vertices (4,0) & (-4,0) vertices (4,0) & (-4,0) foci (5,0) & (-5,0) foci (5,0) & (-5,0) asymptotes y = + 3 / 4 x asymptotes y = + 3 / 4 x