Pre-Calculussssssssssssssss (Part 3).pdf

This is the solar eclipse.

1. The Ellipse
Definition, Parts, Equation, and Graphing

2. ellipse Definition of an Ellipse
1
Set of all points where the sum of distances from two fixed points (foci) is constant
2
Oval-shaped conic section
3
Formed when a plane intersects a cone at an angle not perpendicular to the base

3. center_focus_strongCenter
timeline
Intersection Point
Where major and minor axes intersect
my_location
Reference Point
From which all other parts are
measured
pin_drop
Notation
In standard form, denoted by:
(h, k)
Center (h, k)

4. horizontal_distributeMajor Axis
straighten
Longest Diameter
The longest diameter of the ellipse
timeline
Passes Through
Center, major vertices, and foci
calculate
Length
Total length:
2a
Where a = distance from center to major
vertex
Major Axis
a a
Center

5. vertical_distributeMinor Axis
straighten
Shortest Diameter
The shortest diameter of the ellipse
swap_horiz
Perpendicular
Perpendicular to the major axis, passes
through the center
calculate
Length
Total length:
2b
Where b = distance from center to
minor vertex
Minor Axis
b
b
Center

6. Major Vertices

Endpoints
Endpoints of the major axis

Distance
Located a units from center
a

Orientation
Coordinates depend on
orientation
Horizontal or Vertical
Horizontal Major Axis
(±a, 0)
Vertical Major Axis
(0, ±a)

7. Minor Vertices

Endpoints
Endpoints of the minor axis

Distance
Located b units from center
b

Orientation
Coordinates depend on
orientation
Horizontal or Vertical
Horizontal Major Axis
(0, ±b)
Vertical Major Axis
(±b, 0)

8. Foci

Fixed Points
Two fixed points inside the
ellipse along the major axis
=
Constant Sum
Sum of distances from any
point on ellipse to the two foci
is constant

Distance
Each focus is c units from
center
c²=a²−b²
a
Major Axis
Distance from center to major vertex
b
Minor Axis
Distance from center to minor vertex
c
Focus Distance
Distance from center to each focus

9. center_focus_strongEllipse with Center at the Origin
functionsStandard Equations
Horizontal Major Axis
x²
a²
+
y²
b²
= 1
Vertical Major Axis
x²
b²
+
y²
a²
= 1
placeKey Points Formulas
adjustMajor Vertices
Horizontal: (±a, 0)
Vertical: (0, ±a)
radio_button_uncheckedMinor Vertices
Horizontal: (0, ±b)
Vertical: (±b, 0)
lensFoci
Horizontal: (±c, 0)
Vertical: (0, ±c)
calculateDistance Calculation
c=√(a² - b²)

10. center_focus_weakEllipse with Center not at the Origin
functionsStandard Equations
Horizontal Major Axis
(x - h)²
a²
+
(y - k)²
b²
= 1
Vertical Major Axis
(x - h)²
b²
+
(y - k)²
a²
= 1
placeKey Points Formulas
adjustMajor Vertices
Horizontal: (h ± a, k)
Vertical: (h, k ± a)
radio_button_uncheckedMinor Vertices
Horizontal: (h, k ± b)
Vertical: (h ± b, k)
lensFoci
Horizontal: (h ± c, k)
Vertical: (h, k ± c)
calculateDistance Calculation
c=√(a² - b²)

11. screen_rotationOrientation of the Ellipse
lightbulb Orientation determined by position of a²
horizontal_distribute
Horizontal Major Axis
(x² / a²) + (y² / b²) = 1
When a² is under x
vertical_distribute
Vertical Major Axis
(x² / b²) + (y² / a²) = 1
When a² is under y
This determines the shape's direction and affects coordinates of vertices and foci

12. checklistSteps to Analyze or Graph an Ellipse
1Identify the center (h, k)
center_focus_strong
2Determine a² and b²
straighten
3Find c using c² = a² - b²
calculate
4Plot the center, major and minor vertices, and foci
place
5Draw the ellipse symmetrically using these key points
gesture

13. functionsExample Problem
(x - 1)²
25
+
(y + 2)²
9
= 1
Find:
center_focus_strong
Centerscreen_rotation
Orientationstraighten
a, b, c
adjust
Major Verticesradio_button_unchecked
Minor Verticeslens
Foci
gesture
Sketch the ellipse

14. psychologyTry This!
(x + 3)²
16
+
(y - 1)²
4
= 1
Identify and plot all key parts:
center_focus_strong
Center
screen_rotation
Orientation
straighten
a, b, c
adjust
Major Vertices
radio_button_unchecked
Minor Vertices
lens
Foci

15. assignmentMore Practice Problems
lightbulbFor each ellipse, identify:
center_focus_strongCenterscreen_rotationOrientationstraightena, b, cadjustMajor Verticesradio_button_uncheckedMinor VerticeslensFoci
1
x²
36
+
y²
16
= 1 2
(x - 2)²
49
+
(y + 1)²
25
= 1
3
(x + 4)²
25
+
(y - 3)²
9
= 1 4
(x - 5)²
4
+
(y + 2)²
16
= 1
5
(x + 1)²
100
+
(y - 4)²
36
= 1 6
(x - 3)²
64
+
(y + 2)²
36
= 1
7
(x + 6)²
49
+
(y - 5)²
9
= 1 8
(x - 2)²
9
+
(y + 1)²
25
= 1