mat151: chapter 7, pg 1 7.1. parabolas mat 151 chapter 7
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mat151: Chapter 7, Pg 1
7.1. Parabolas7.1. Parabolas
Mat 151
Chapter 7
mat151: Chapter 7, Pg 2
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
mat151: Chapter 7, Pg 3
7.1. PARABOLA7.1. PARABOLA
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
mat151: Chapter 7, Pg 4
7.1. PARABOLA7.1. 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
mat151: Chapter 7, Pg 5
7.1. PARABOLA7.1. PARABOLA
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
mat151: Chapter 7, Pg 6
7.1. PARABOLA7.1. PARABOLA
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
mat151: Chapter 7, Pg 7
7.1. PARABOLA7.1. PARABOLA
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
mat151: Chapter 7, Pg 8
7.1. PARABOLA7.1. PARABOLA
Graph the horizontal parabola:
322 2 yyx
2)1(3 2 yx
mat151: Chapter 7, Pg 9
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.
mat151: Chapter 7, Pg 10
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.
mat151: Chapter 7, Pg 11
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
mat151: Chapter 7, Pg 12
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
mat151: Chapter 7, Pg 13
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)
mat151: Chapter 7, Pg 14
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
mat151: Chapter 7, Pg 15
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
mat151: Chapter 7, Pg 16
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)
mat151: Chapter 7, Pg 17
7.2. ELLIPSE7.2. ELLIPSE
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
mat151: Chapter 7, Pg 18
7.2. ELLIPSE7.2. ELLIPSE
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
mat151: Chapter 7, Pg 19
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
mat151: Chapter 7, Pg 20
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)
mat151: Chapter 7, Pg 21
y
x
V2= (0, a)
V1= (0, -a)
(b, 0)(-b, 0)
F2 = (0, c)
F1= (0, -c)
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