10.2 Notes Precalculus ~ Parabolas Name A parabola is the ...

10.2 Notes Precalculus ~ Parabolas Name ______________________________

A parabola is the set of all points (x,y) in a plane that are equidistant from a fixed line (directrix) and a fixed

point (focus) not of the line.

Standard form:  x  h2  4 p  y  k  Standard form:  y  k 2  4 p  x  h
Vertex: h, k  Vertex: h, k 

Axis of symmetry(vertical): x  h Axis of symmetry (horizontal): y  k

Focus: h, k  p Focus: h  p, k 

Directrix: y  k  p Directrix: x  h  p
If p  0, parabola opens up If p  0, parabola opens right
If p  0, parabola opens down If p  0, parabola opens left

-Vertex is the midpoint between the focus and directrix.
-Axis of symmetry is the line passing through the focus and vertex and is perpendicular to the directrix
-Focus is “p” units away from vertex always inside the curve.
-d1 represents distance from focus to point on parabola, d2 represents distance from focus to point on parabola
- d1 = d2 for any point chosen on parabola
-the latus rectum is a segment through the focus that is perpendicular to the axis of symmetry

Ex1) Find standard form of a parabola with vertex at the origin and focus at (2, 0).

Ex2) Find standard form of a parabola with vertex (2,1) and focus (2, 4).

Ex 3) Find standard form of a parabola with vertex at the origin and a directrix x = 2.

Ex 4) Find standard form, vertex, focus, and directrix of the parabola: y   1 x2  x  1
2 2

Ex 5) Change the equation of the parabola so that its graph is the lower half of the parabola:  y 12  2 x  4

Ex 6) The revenue R generated by the sale of x units of a digital camera is  x 1352   5  R  25,515 .

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Approximate the number of sales that will maximize revenue.

Ex 7) Find the standard form of the equation with vertex at the origin and:

A) Passes through (-3, 6) with a B) Focus at (0, -2)
horizontal axis of symmetry.

Axis of Tangent
symmetry line

Reflective Property of a Parabola focus P
The tangent line to a parabola at a point P makes equal angles with
the following two lines:

1. The line passing through P and the focus
2. The axis of symmetry for the parabola at point R
The distance from P to focus and R to focus are equal.

R

Ex 8) Find the equation of the tangent line to a parabola given y  x2 at the point (1, 1).
Vertex ( _____ , ______ )
Focus ( ______ , ______ )
Distance = ___________ (focus to given point P)
2nd point R ( _____ , ______ ) (distance from focus)

PreCalculus Opener 10.2
1. Find the standard form, vertex, focus, and directrix of the parabola. y2  6y 8x 17  0
Standard Form: _____________________________
Vertex:
Focus:
Directrix:

2. Find the standard form of the parabola given a vertex at (1, 5) and directrix: x = 3

3. Find the standard form of the parabola given vertex (-2,1) and focus at (3,1).

4. Find the standard form of the equation with vertex at the origin, horizontal axis and goes through the point
(-2,6)

5. Find the equation of the tangent line to a parabola given y  x2 at the point (2, 4).
Vertex ( _____ , ______ )
Focus ( ______ , ______ )

10.3 Notes Precalculus ~ Ellipses Name ______________________________

An ellipse is the set of all points (x,y) in a plane, the sum of whose distances from two distinct points (foci) is a

constant.

(x,y)

minor

axis

vertex vertex

focus focus center major

axis

d1  d2  Constant Center (h, k)

major axis = 2a
minor axis = 2b

A segment through each foci perpendicular to the major axis is the latus rectum. Four additional point of the
ellipse are the endpoints of the two latera recta. These are helpful when sketching an ellipse.

The length of each latus rectum = 2b2
a

The standard form of the equation of an ellipse centered at (h,k) and having a major axis that is

Horizontal: Vertical:

 x  h2  y  k 2  x  h2  y  k 2

 1  1
a2 b2 b2 a2

*In both of these cases, the foci lie on the major axis, c units ba
from the center, with c2  a2  b2

c
a foci

Eccentricity :( e  c )-measures the “ovalness” of an ellipse. b
a

0  e 1 for every ellipse. For an ellipse that is nearly circular, the foci are close to the center and eccentricity

is very small. However the closer e is to one (foci are close to vertices), the more elongated an ellipse becomes.

Ex 1) Graph the equation using the four vertices, and four endpoints of latera recta.  x 12   y  22  1

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Ex2) Find the standard form of the equation of the ellipse having foci (0, 1) and (4, 1) and a major axis of length 6.
What is the Center?
What is the elongation?
c = _____, a = _____, b = _______

Ex3) Find the standard form of the ellipse and answer all questions, given the major vertices (3, 8) & (3, -2)
and the minor vertices (0,3) & (6,3)

What is the elongation?_______________________
Center: (______, ______)
a = _______, b = _______, c = _______
Foci: (______, __________) & (______, ___________)
Eccentricity: _____________
Length of Latus Rectum: __________
Endpts of Latera Recta: (______, ______) (______, ______) (______, ______) (______, ______)

Ex 4) Find the standard form of the ellipse and answer all questions: 4x2  y2 8x  4y 8  0

What is the elongation?_______________________
Center: (______, ______)
a = _______, b = _______, c = _______
Major vertices: (______, ______) & (______, ______)
Minor vertices: (______, ______) & (______, ______)
Foci: (______, __________) & (______, ___________)
Eccentricity: ________Length of Latus Rectum: _________
Endpts of Latera Recta: (______, ______) & (______, ______) & (______, ______) & (______, ______)

Ex 5) Write the equation of the ellipse in standard form. Center (2, -1); vertex: (2, ½); minor axis of length 2.

Ellipse --------major axis-------- --minor axis--

horizontal elongation Vertical elongation

Note:
1.)Needs a center(h,k), “+” equation
2.) a2 is always the bigger number; 2a=major axis

b2 is always the smaller number; 2b = minor axis
3.) Each foci is on the major axis c units from the center. (
4.) Two latera recta, each = (the latera recta are to major axis through each foci)
5.) Eccentricity =

1. 36x2  9y2  216x  36y  36  0  2. Find the standard form of the ellipse with vertices: 0, 5
and foci: 0, 2

Standard Form: ________________________

Elongation: _____________________

Center: ________________________

a = _____ b = _____ c = _____

Vertices:  ____, ____ ,  ____, ____ ,

 ____, ____ , and  ____, ____

Foci:  ______, ______ and  ______, ______

Eccentricity: _________________

Endpoints of Latera Recta:

______, ______ ______, ______
______, ______ ______, ______

3. Endpoints of major axis at (0, -3) and (8, -3).
Endpoints of minor axis at (4,0) and (4,-6).
Write the equation.

Hyperbola

 x  h2  y  k 2  y  k 2  x  h2

 1  1
a2 b2 a2 b2

Note: Conjugate axis = 2b
1.)“-“ equation Transverse axis = 2a
2.)a2 is always 1st number; b2 is always 2nd number
(connects vertices)
3.) Foci are inside curves located on transverse axis, c units from center.

4.) Two latera recta, each = (the latera recta are to major axis through each foci)

5.) Eccentricity =

6.) asymptotes: if transverse axis is vertical

if transverse axis is horizontal

Ex 1: Ex 2:

10.4 Notes Precalculus ~ Hyperbola
A hyperbola is the set of all points (x,y) in a plane, the difference of whose distances from two distinct fixed
points (foci) is a constant.

 x  h2  y  k 2 1  y  k 2  x  h2
  1
a2 b2 a2 b2

* In both of these cases:
 center (h,k)
 the center is the midpoint of the vertices and the foci.
 The foci and vertices lie on the transverse axis. The vertices are a units from the center and foci

are c units from center, with c2  a2  b2

 Eccentricity e  c , since c > a, e > 1 for every hyperbola
a

 2b2
length each of latus rectum =
a

Ex1) Find the standard form of the equation of the hyperbola having foci (-1,2) and (5,2) and vertices (0,2) and
(4,2).

Ex2) Find the standard form of the hyperbola, graph and answer all questions: 4x2  y2  16
Center: (______, ______)
a = _______, b = _______, c = _______
vertices: (______, ______) & (______, ______)
Foci: (______, __________) & (______, ___________)
Eccentricity:

Ex3) Find the standard form of the hyperbola, graph and answer all questions: 4y2  3x2  8y 16  0

Center: (______, ______)
a = _______, b = _______, c = _______
vertices: (______, ______) & (______, ______)
Foci: (______, __________) & (______, ___________)
Equation of asymptotes:_____________________________
________________________________________________
Length of Latus rectum: __________
Endpts of Latera Recta: (______, ______) & (______, ______)
& (______, ______) & (______, ______)

Ex 4) Find the standard form of the equation of the hyperbola with vertices at (3, 0) and (3, 4);
Asymptotes: y  2 x and y   2 x  4

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Classifying Conics from general form: Ax2  Cy2  Dx  Ey  F  0

1. Circle: A = C
2. Parabola: AC = 0 (Either A= 0 or C = 0, but not both)
3. Ellipse: AC >0 (A and C have the same signs)
4. Hyperbola: AC < 0 (A and C have opposite signs)

Ex 5) Classify the graph of each equation: B. 4x2  y2  8x  6y  4  0
A. 4x2  9x  y  5  0

C. 2x2  4y2  4x 12y  0 D. 2x2  2y2  8x 12y  2  0

PreCalculus 10.2-10.4 Conics
1. Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.

A) B)
C) D)

2. Find the vertices, foci, and equations of the asymptotes of the hyperbola.
Vertices: (_____, _____) &(______, ______)
Foci: (_____, _____) &(______, ______)
Asymptotes: ___________________________
Endpoints of latera recta: (_____, _____) &(______, ______)
(_____, _____) &(______, ______)

3. Find the vertices, foci, and equations of the asymptotes of the hyperbola.
Standard Form: ___________________________
Vertices: (_____, _____) &(______, ______)
Foci: (_____, _____) &(______, ______)
Asymptotes: ___________________________

4. Find standard form of an ellipse with foci: (8, 2), (4,2); Major axis of length 6.

5. Erik rides her bike through a tunnel shaped like the top half of an ellipse. The tunnel is 6.7 meters wide and
2.4 meters high. On her bike, the top of Erik’s helmet is 1.8 meters above the ground. If she were to ride
through the tunnel 2.1 meters from the center, would her helmet miss the ceiling? If so, how much room above
his head?

6. In a factory, a parabolic mirror to be used in a searchlight was placed on the floor. It measured 50 cm tall
and 100 cm wide. Find an equation of the parabola with its vertex at the origin.

50cm
100cm