Geo: Unit 5: Conic Sections

props of parabolas

The directed distance from the vertex to the directrix is −f . f≠0 and a≠0 . f>0 and a>0 if the parabola opens right or up. f<0 and a<0 if the parabola opens left or down. x−h=a(y−k)2 or y−k=a(x−h)2 equation in graphing form

Which equation represents a parabola that has a focus of (0,0) and a directrix of y = 8?

V=(0+0/2,0+8/2)V=(0/2,8/2) V=(0,4) f=0−4 f=−4(x−h)/2=4f(y−k)(x−0)2=4(−4)(y−4) x2=−16(y−4)

conjugate binomials

binomials that are the same except for the sign between the terms (a+b) and (a−b)

(x−1)2+(y+3)2=5^2

center= (1,−3) radius r is 5

Write the equation in graphing form. Then identify the focus and directrix of the parabola.y = 12x2−4x+10

complete the square: y = 12x2−4x+10 y−10=12x2−4x y−10=12(x2−8x) (b2)2=(−82)2=16 y−10+8=12(x2−8x+16) y−2=12(x−4)2

ellipse

conic section formed when a plane intersects only one nappe, not parallel to the generating line. A circle can be considered a special kind of ellipse.

parabola

conic section formed when a plane intersects only one nappe, parallel to the generating line.

circleee

conic section formed when a plane intersects only one nappe, perpendicular to the axis.

circle

diameter= d= 2r circumference= C= 2pi r area= A= pi r2

focal length f

directed distance from vertex (h,k) to the focus of the parabola -focal length= 1/4a -a= 1/4f

how to make a perfect square trinomial

divide the x term by two and square the result x^2 +8x + ? 8/2= 4 4^2= 16

linear function

f(x)=Ax+By+C first degree equation

quadratic function

f(x)=ax2+bx+c second degree equation -forms parabola that either opens up or down

Directrix

fixed line that all points on a parabola are equidistant from

focus

fixed point where all points on parabola are equidistant from

parabolaaa

formed when a plane intersects the top or bottom half of a double-napped cone parallel to the generating line, but does not intersect the vertex

degenerate conic sections

formed when a plane intersects the vertex of the double-napped cone. There are three types of degenerate conic sections.

vertexx

halfway between the focus and the point where the axis of symmetry intersects the directrix -directrix is horizontal or vertical y=c or x=c, where c is a constant.

axis of symmetry

line that passes through the focus perpendicular to the directrix -equation of vertical or horizontal line

how to change general form into standard form

multiply the equation out separating both sides and then make 2 perfect square trinomials -combine the two terms simplify

nappe

one of the two equal pieces of a double cone when the cones are connected at the vertex by a plane perpendicular to the axis.

result of squaring a binomial

perfect square trinomial -not a perfect square trinomial if it doesn't become a square of a binomial (x + 3)^2

degenerate parabola

plane contains the generating lin

degenerate hyperbola

plane intersects both nappes through the vertex

degenerate ellipse

plane intersects only the vertex

vertex

point where lines intersect on the cone

completing the square

process of transforming an expression of the form x2 + bx into a perfect square trinomial by adding the term (b/2)2 to it

how to determine the center and raidus of circle with equation

rewrite equation as the distance formula pull out the necessary x, y, and r values

circle def

set of all points in a plane that are a fixed distance, r, called the radius, from a given point called the center. In a coordinate plane, a circle is a set of points (x,y). For any circle, r has the same value no matter where (x,y) is on the circle.

vertex equals f

subtract x and y of the vertex to find focus point

y term squared

the axis of symmetry is horizontal with the parabola opening left or right (y−k)2=4f(x−h) or a(y−k)2=x−h

x term squared

the axis of symmetry is vertical with the parabola opening up or down. x−h)2=4f(y−k) or a(x−h)2=y−k

conic section

two-dimensional graph that can be formed by the intersection of a plane with a double-napped cone.

equation of circle in standard form

with center (h,k) and radius r, is (x−h)2+(y−k)2=r2

general form of the equation of a circle

x2+y2+Ax+By+C=0

standard form

x2+y2=r2

graphing formm

y+3=2(x+2)2

product of conjugate binomials

(a+b)(a−b)=a2−ab+ab−b2 =a2−b2 a2−b2= difference of two squares

vertex location

(h,k) a(x−h)2=y−k (y−k)2=4f(x−h)

quadratic formula

-b±[√b²-4ac]/2a

how to rewrite equation of circle in standard form

1. move constant to right side 2. group terms 3. solve out the square for each binomial, add the same number to the other side 4. add the right side 5. solve everthing

how to derive equation of circl

1. use distance formula to set up equation plugging in the center for x1 and y1 and put it equal to the radius 2. sove until you get a nice equation, no double parenthesis, no negative negative

standard form of the equation of a circle

A circle with center (h,k) and radius r has this equation: (x−h)2+(y−k)2=r2

horizontal axis of symmetry

Equation in graphing form: x−h=a(y−k)2

parabole with vertical axis of symmetry

Equation in graphing form: y−k=a(x−h)2

Find the vertex of the parabola. (y−1)^2 = m4(x−3)

Find the vertex of the parabola. (y−1)2=4(x−3) Identify the vertex. (3,1) The vertex is represented by (h,k) in the equation (y−k)2=4f(x−h).

Determine the vertex, focus, and directrix of the parabola. x2=−4y

Identify the vertex. (0,0) Identify the focus. 4f=−4 4f/4=−4/4 f=−1 The focus is defined by (h,k+f). (h,k+f) (0,0+(−1)) (0,−1) Identify the directrix y=k−f y=0−(−1) y=1

Find the focus of the parabola. 18(y+2)2=x−5

Identify the vertex. (5,−2) a(y−k)2=x−h Determine the focus. f=1/4a f=1/4(1/8) f=1/(12) f=2

double-napped cone

One line, called the generating line, intersects and revolves around another line, called the axis. The axis is stationary, and the two lines cannot be perpendicular.