Mth 162 Pre Cal 2 Unit 4

Front 1

Standard form of an equation of a circle

Back 1

Given a circle centered at (h,k) with radius r, the standard form of an equation of the circle is given by (x-h)squared + (y-k)squared=r squared for r > 0

Front 2

Ellipse

Back 2

Set of all points (x,y) in a plane such that the sum of the distances between (x,y) and two fixed points is a constant. The fixed points are called the foci (plural of focus) of the ellipse

Front 3

Vertices

Back 3

The line through the foci that intersects the ellipse at two points

Front 4

Major axis

Back 4

The line segment with endpoints at the vertices

Front 5

Center of the ellipse

Back 5

Midpoint of the major axis

Front 6

Minor axis

Back 6

Line segment perpendicular to the major axis and passing through the center of the ellipse with endpoints on the ellipse

Front 7

Standard form of an equation of an ellipse centered at the origin. Xaxis

Back 7

Equation: (xsquared/asquared)+(ysquared/bsquared)=1Center: (0,0)Foci (note: csquared=asquared-bsquared): (c,0) and (-c,0)Vertices: Endpoints: major axis (a,0) and (-a,0)Endpoints: minor axis (0,b) and (0,-b)

Equation: (xsquared/asquared)+(ysquared/bsquared)=1Center: (0,0)Foci (note: csquared=asquared-bsquared): (c,0) and (-c,0)Vertices: Endpoints: major axis (a,0) and (-a,0)Endpoints: minor axis (0,b) and (0,-b)

Front 8

Standard form of an equation of an ellipse centered at the origin. Y axis

Back 8

Equation: (xsquared/bsquared)+(ysquared/asquared)=1Center: (0,0)Foci (note: csquared=asquared-bsquared): (0,c) and (0,-c)

Vertices: Endpoints: major axis (0,a) and (0,-a)Endpoints: minor axis (b,0) and (-b,0)

Front 9

Standard form of an equation of an ellipse centered at (h,k) major axis horizontal

Back 9

Equation:((X-h)squared/asquared)+((y-k)squared/bsquared)=1

Center: (h,k)

Foci note csquared=asquared-bsquared: (h+c,k) and (h-c,k)

Vertices: (h+a,k) and (h-a,k)

Endpoints minor axis: (h,k+b) and (h,k-b)

Front 10

Standard form of an equation of an ellipse centered at (h,k) major axis vertical

Back 10

Equation:((X-h)squared/bsquared)+((y-k)squared/asquared)=1Center: (h,k)Foci note csquared=asquared-bsquared: (h,k+c) and (h,k-c)Vertices: (h,k+a) and (h,k-a)Endpoints minor axis: (h+b,k) and (h-b,k)

Front 11

Eccentricity of an ellipse

Back 11

For an ellipse defined by the standard form of an equation of an ellipse centered at (h,k), the eccentricity e is given by e=c/a where a>b>0, c>0, and csquared=asquared-bsquared

Note: the eccentricity is a number where 0<e<1

Front 12

Hyperbola

Back 12

Set of all points (x,y) in a plane such that the difference in distances between (x,y) and two fixed points (foci) is a positive constant

Front 13

Transverse axis

Back 13

The line segment between the vertices of a hyperbola

Front 14

Standard form of an equation of a hyperbola centered at the origin. Transverse axis: x axis

Back 14

Equation: (xsquared/asquared) -(ysquared/bsquared)

Center: (0,0)

Foci (csquared=asquared+bsquared): (c,0) and (-c,0)

Vertices: (a,0) and (-a,0)

Asymptotes: y=(b/a)x and y=(-b/a)x

Front 15

Standard form of an equation of a hyperbola centered at the origin. Transverse axis: y axis

Back 15

Equation: (ysquared/asquared) -(xsquared/bsquared)Center: (0,0)Foci (csquared=asquared+bsquared): (0,c) and (0,-c)Vertices: (0,a) and (0,-a)Asymptotes: y=(a/b)x and y=(-a/b)x

Front 16

Graphing a hyperbola

Back 16

Step1: identify the center and vertices

Step 2: draw the reference rectangle centered at the center of the hyperbola, with dimensions 2a and 2b

Step 3: draw the asymptotes through the opposite corners of the rectangle

Step 4: sketch each branch of the hyperbola starting at the vertices and approaching the asymptotes

Front 17

Standard forms of an equation of a hyperbola centered at (h,k). Transverse axis horizontal

Back 17

Equation: (((x-h)squared)/asquared)-(((y-k)squared)/bsquared)=1

Center: (h,k)

Foci (Note: csquared=asquared+bsquared): (h+c,k) and (h-c,k)

Vertices: (h+a,k) and (h-a,k)

Asymptotes: y-k=+-(b/a)(x-h)

Front 18

Standard forms of an equation of a hyperbola centered at (h,k). Transverse axis vertical

Back 18

Equation: (((y-k)squared)/asquared)-(((x-h)squared)/bsquared)=1Center: (h,k)Foci (Note: csquared=asquared+bsquared): (h,k+c) and (h,k-c)Vertices: (h,k+a) and (h,k-a)Asymptotes: y-k=+-(a/b)(x-h)

Front 19

Eccentricity of a hyperbola

Back 19

For a hyperbola in standard form the eccentricity e is given by e = c/a where a>0, c>0, and csquared=asquared+bsquared. Note: the eccentricity of a hyperbola is a number greater than 1

Front 20

Parabola

Back 20

Set of all points in a plane that are equidistant from a fixed line (called the directrix) and a fixed point called the focus

Front 21

Standrad form of an equation of a parabola with vertex at the origin. Y axis is axis of symmetry

Back 21

Equation: xsquared =4py

Vertex: (0,0)

Focus: (0,p)

Directrix: y=-p

Front 22

Standrad form of an equation of a parabola with vertex at the origin. X axis is axis of symmetry

Back 22

Equation: ysquared =4py

Vertex: (0,0)

Focus: (p,0)

Directrix: x=-p

Front 23

Standard form of an equation of a parabola with vertex (h,k) vertical axis of symmetry

Back 23

Equation: (x-h)squared=4p(y-k)

Vertex: (h,k)

Focus: (h,k+p)

Directrix: y=k-p

Axis of symmetry: x=h

Front 24

Standard form of an equation of a parabola with vertex (h,k) horizontal axis of symmetry

Back 24

Equation: (y-h)squared=4p(x-k)

Vertex: (h,k)

Focus: (h+p,k)

Directrix: x=h-p

Axis of symmetry: y=k