7.1. parabolas

mat151: Chapter 7, Pg 1

7.1. Parabolas7.1. Parabolas

Mat 151

Chapter 7


7.1. PARABOLA7.1. PARABOLA

Vertical Parabola – x is squared but not y

khxay 2)(

1. Vertex (h, k)2. If a > 0 Opens UP3. If a < 0 Opens DOWN4. If a > 1 then parabola is SKINNY ( or a < - 1)5. If - 1 < a < 1 parabola is FAT



For every parabola find:

2)4( xy

1. Find the vertex2. Find the y and x intercepts3. Graph the parabola4. Find the axis of symmetry for

parabola



Graph parabola find:2)4( xy

-4

-2

0

2

4

-6 -4 -2 0 2 4 6

1. Vertex @ (4,0)2. X - intercept @ (4,0)3. Y - intercept @ (0,16)4. Parabola opens up5. Axis of symmetry for

parabola is vertical line x = 4



Graph parabola find:

2)3( 2 xy 1. Vertex @ (3, - 2)2. X - intercept @ (4.41,0) and

(1.59,0)3. Y - intercept @ (0,7)4. Parabola opens up5. Axis of symmetry for

parabola is vertical line x = 3

-4

-2

0

2

4

-6 -4 -2 0 2 4 6



Horizontal Parabola – y is squared but not x

hkyax 2)(

1. Vertex (h, k)2. If a > 0 Opens to the right3. If a < 0 Opens to the left4. If a > 1 then parabola is SKINNY ( or a < - 1)5. If - 1 < a < 1 parabola is FAT



Graph parabola find:

22 yx 1. Vertex @ (2,0)2. X - intercept @ (2,0)3. Y - intercept – No y intercept4. Parabola opens to the right5. Axis of symmetry for parabola

is horizontal line y = 0

-2

-1

0

1

2

-4 -3 -2 -1 0 1 2 3 4



Graph the horizontal parabola:

322 2 yyx

2)1(3 2 yx


7.1. Application of 7.1. Application of PARABOLAPARABOLA

If an object is thrown upward with initial velocity of 32 ft/sec, then its height after t seconds is:

21632 tth 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.


7.1. Application of 7.1. Application of PARABOLAPARABOLA

The revenue received from selling x stereos is given by the formula:

100805.0 2 xxR 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 - 100 is information that has the

maximum height and also half of total time in air.


An equation of the ellipse with center at (0, 0) and foci at (- c, 0) and (c, 0) is:

2222

2

2

2

and 0 where1 cabbab

y

a

x

Because a > b the major axis is the x-axis

The vertices are at (-a, 0) and (a, 0).

7.2 EQUATION OF ELLIPSE


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)

F2F1 V2V1

Major Axis

Minor Axis

ELLIPSE


b a c2 2 2

x

a

y

b

2

2

2

2 1

y

x

F2=(c, 0)F1=(-c, 0)

V2=(a, 0)

(0, b)

(0, -b)

GRAPH OF ELLIPSE

V1=(-a, 0)


Ellipse with Major Axis Parallel to the x-Axis where a > b and b2 = a2 - c2.

Equation Center Foci Vertices

x h

a

y k

b

2

2

2

21

(h, k) (h + c, k) (h + a, k)

y

(h - a, k) (h + a, k)(h, k)

(h - c, k) (h + c, k)

x

Major axis


Ellipse with Major Axis Parallel to the y-Axis where a > b and b2 = a2 - c2.

Equation Center Foci Vertices

x h

b

y k

a

2

2

2

21

(h, k) (h, k + c) (h, k + a)

y (h, k + a)

x(h, k - a)

(h, k)

(h, k + c)

(h, k - c)

Major axis


7.2. ELLIPSE7.2. ELLIPSE

Graph the ellipse:

153 2

2

2

2

yx 1. Center @ (0,0)

2. X - intercepts @ (- 3,0) and (3,0)

3. Y - intercepts @ (0,- 5) and (0,5)

4. a = 55. b = 3

x

(0, 5)

(0, -5)

(3, 0)(-3, 0)



Graph the ellipse:

1916

22

yx 1. Center @ (0,0)

2. a = 43. b = 34. X - intercepts @ (- 4,0) and

(4,0)5. Y - intercepts @ (0,- 3) and

(0,3)x

(0, 4)(0, -4)

(3, 0)

(-3, 0)y



Graph the ellipse:

19

)1(

16

)2( 22

yx 1. Center @ (-2 , - 1)

2. Horizontal axis is a = 43. Vertical axis is b = 3

From the center:- Go 3 units UP- Go 3 units DOWN- Go 4 units RIGHT- Go 4 units LEFTConnect four points

x

(-2, -1)

(-2, -4)

(-2, 2)

(-6, -1) (2, -1)

y


7.2. Application of ELLIPSE7.2. 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?

From the graph:2a = 20 a = 10 ftb = 15 fth = 0k = 0

x

(-6, -1)

y

2a = 20

b = 15


7.2. Application of ELLIPSE7.2. 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.

h

(-6, -1)

2a = 20

b = 15

12ft

12

2

2

2

b

y

a

x1

1510 2

2

2

2

yx

If we consider ellipse centered @ (0,0) then a = 10 and b = 15

If we substitute x = 6:

11510

62

2

2

2

y

fthy 12 Height of the truck

(-6,y) (6,y)


y

x

V2= (0, a)

V1= (0, -a)

(b, 0)(-b, 0)

F2 = (0, c)

F1= (0, -c)

7.1. parabolas

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