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.