1. Mat 151
Chapter 7
7.1. Parabolas

2. 7.1. PARABOLA Vertical Parabola – x is squared but not y Vertex (h, k)
If a > 0 Opens UP
If a < 0 Opens DOWN
If a > 1 then parabola is SKINNY ( or a < - 1)
If < a < 1 parabola is FAT

3. 7.1. PARABOLA For every parabola find: Find the vertex
Find the y and x intercepts
Graph the parabola
Find the axis of symmetry for parabola

4. 7.1. PARABOLA Graph parabola find: Vertex @ (4,0)
X - (4,0)
Y - (0,16)
Parabola opens up
Axis of symmetry for parabola is vertical line x = 4

5. 7.1. PARABOLA Graph parabola find: Vertex @ (3, - 2)
X - (4.41,0) and (1.59,0)
Y - (0,7)
Parabola opens up
Axis of symmetry for parabola is vertical line x = 3

6. 7.1. PARABOLA Horizontal Parabola – y is squared but not x
Vertex (h, k)
If a > 0 Opens to the right
If a < 0 Opens to the left
If a > 1 then parabola is SKINNY ( or a < - 1)
If < a < 1 parabola is FAT

7. 7.1. PARABOLA Graph parabola find: Vertex @ (2,0)
X - (2,0)
Y - intercept – No y intercept
Parabola opens to the right
Axis of symmetry for parabola is horizontal line y = 0

8. 7.1. PARABOLA
Graph the horizontal parabola:

9. 7.1. Application of PARABOLA
If an object is thrown upward with initial velocity of 32 ft/sec, then its height after t seconds is:
Find the maximum height attained by the object.
Find the total time in air.
HINT:
The vertex of parabola h = -16t2 + 32t is information that has the maximum height and also half of total time in air.

10. 7.1. Application of PARABOLA
The revenue received from selling x stereos is given by the formula:
Find how many stereos must be sold to obtain the maximum revenue?
Find the maximum revenue.
HINT:
The vertex of parabola R = -0.5x2 + 80x is information that has the maximum height and also half of total time in air.

11. 7.2 EQUATION OF ELLIPSE
An equation of the ellipse with center at (0, 0) and foci at (- c, 0) and (c, 0) is:
Because a > b the major axis is the x-axis
The vertices are at (-a, 0) and (a, 0).

12. ELLIPSE P = (x, y) F2 F1 V2 V1 Major Axis Minor Axis
An ellipse is the collection of points in the plane the sum of whose distances from two fixed points, called the foci, is a constant. y x
P = (x, y) F2 F1 V2 V1
Major Axis
Minor Axis

13. GRAPH OF ELLIPSE y F2=(c, 0) F1=(-c, 0) (0, b) x V2=(a, 0) V1=(-a, 0)

14. Ellipse with Major Axis Parallel to the x-Axis where a > b and b2 = a2 - c2.y
(h - a, k)
(h + a, k)
(h, k)
(h - c, k)
(h + c, k) x
Major axis

15. Ellipse with Major Axis Parallel to the y-Axis where a > b and b2 = a2 - c2.
(h, k + a) x
(h, k - a)
(h, k)
(h, k + c)
(h, k - c)
Major axis

16. 7.2. ELLIPSE Graph the ellipse: x Center @ (0,0)
X - (- 3,0) and (3,0)
Y - (0,- 5) and (0,5)
a = 5
b = 3
(0, 5)
(-3, 0)
(3, 0) x
(0, -5)

17. 7.2. ELLIPSE Graph the ellipse: y x Center @ (0,0) a = 4 b = 3
X - (- 4,0) and (4,0)
Y - (0,- 3) and (0,3) y
(-3, 0) x
(0, -4)
(0, 4)
(3, 0)

18. 7.2. ELLIPSE Graph the ellipse: y x Center @ (-2 , - 1)
Horizontal axis is a = 4
Vertical axis is b = 3 x
(-2, -1)
(-2, -4)
(-2, 2)
(-6, -1)
(2, -1) y
From the center:
- Go 3 units UP
- Go 3 units DOWN
- Go 4 units RIGHT
- Go 4 units LEFT
Connect four points

19. 7.2. Application of ELLIPSE
A one way road passes an overpass in the form of half of an ellipse, 15 ft high at the center and 20 ft wide. Assuming a truck is 12 ft wide, what is the tallest truck that can pass under the overpass? x
(-6, -1) y
2a = 20
b = 15
From the graph:
2a = 20 a = 10 ft
b = 15 ft
h = 0
k = 0

20. 7.2. Application of ELLIPSE
Solution: The height of truck is x
If we have equation for ellipse, and substitute x = - 6 or x = 6, we will find the y that represent the height of truck.
(-6,y)
(6,y)
b = 15 h
(-6, -1)
12ft
2a = 20
If we consider ellipse (0,0) then a = 10 and b = 15
If we substitute x = 6:
Height of the truck

21. y x
V2= (0, a)
V1= (0, -a)
(b, 0)
(-b, 0)
F2 = (0, c)
F1= (0, -c)