1. Solve the system
x−y =4
3x2 − x + y = 8
1. Solve the system
y =x−4
line
3x2 − x + (x − 4) = 8
3x2 − −4 = 8
3x2 = 12
x2 = 4
x = ±2
If x = 2:
y = -2
If x = -2
y = -6
(2, -2)
(-2, -6)
Parabola
2. Solve the system
x2 − y 2 = 16
4y 2 + 4x = 16
2. Solve the system
y2 = 4 − x
parabola
x2 − y 2 = 16
hyperbola
x2 − (4 − x) = 16
x2 + x − 20 = 0
(x + 5)(x − 4) = 0
x = -5, x = 4
If x = -5:
y = ±3
if x = 4,
y=0
(4, 0)
(-5, 3)
(-5, -3)
3. Solve the system
x2 + y 2 + 2y = 3
x2 − y = 5
3. Solve the system
Elimination
x2 + y 2 + 2y = 3
Circle
−x2 + y = −5
Parbola
2
y + 3y + 2 = 0
(y+1)(y+2) = 0
y = -2
y = -1
If y = -1:
x = ±2
√
If√y = -2 √
x=± 3
( 3, −2)(− 3, −2)(2, −1)(−2, −1)
4. Find the equation
1. A parabola with vertical axis of symmetry,
vertex (3, 2) that√contains the point
√ (0, 4)
2.Ellipse: foci (2- 7, -1) & ( 2+ 7, -1)
& co-vertices (2, -4) & (2, 2)
3. Hyperbola with vertices (5, -3) and
1
(-7, -3) and asymptote slope of ±
2
4. Find the equation
9
1.(x − 3)2 = (y − 2)
4
.
(x − 2)2 (y + 1)2
2.
+
=1
16
9
.
(x + 1)2 (y + 3)2
3.
−
=1
16
4
5. Word Problem
The main cables of a suspension bridge are
20 meters above the road at the towers and
4 meters above the road at the center. The
road is 80 meters long. The main cables
hang in the shape of a parabola. Find the
equation of the parabola. Then, determine
how high the main cable is 20 meters from
the center.
5 Word Problem
Vertex (0, 4)
opens up
contains the points (40, 20) and (-40, 20)
(x)2 = 4p(y − 4)
(40)2 = 4p(16)
1600 = 64p
p = 25
x2 = 100(y − 4)
When x = 20 the height is 44 ft
OR Vertex (40, 4)
opens up
6. Word Problem
The Statuary Hall in the United States
Capitol is elliptical. It measures 46 feet wide
and 96 feet long. If a person is standing at
one focus, her whisper can be heard by a
person standing at the other focus. How far
apart are the two people?
6 Word Problem
(x)2 (y)2
+ 2 =1
Center at origin.
482
23
c2 = a2 − b2
c2 = 2304
√ − 529 = 1775
c = 5 71 ≈ 42.12
They are 84.3 ft apart