Multiply and Divide Rational...
Transcript of Multiply and Divide Rational...
HyperbolasLESSON 10.5
Objective
Define hyperbolas and parts of a hyperbola
Graph hyperbolas with center at the origin
Find the equation of a hyperbola
Find the asymptotes of a hyperbola at the origin
Define Hyperbolas
A hyperbola is the set of all points on a plane whose distance from
two fixed focal points (foci) subtract to a constant number.
NOTE: an ellipse adds to be a constant
distance from the foci while a hyperbola
subtracts to be a constant distance.
Define Hyperbolas
The transverse axis is the line containing the foci, the midpoint
of which is the center. The two points of intersection of the
hyperbola and the transverse
axis are the vertices (𝑉1, 𝑉2).
Define Hyperbolas
With a center at the origin, equations for the two
types of hyperbolas follow:
Horizontal Vertical
𝑥2
𝑎2−
𝑦2
𝑏2= 1 or
𝑦2
𝑎2−
𝑥2
𝑏2= 1
Graph Hyperbolas
To graph hyperbolas
1. Identify the center, 𝑎, and 𝑏.
2. Draw a box with 𝑎 being the distance from the
center along the transverse axis
3. Connect opposite corners to create asymptotes
4. Draw hyperbola with respect to asymptotes
Graph Hyperbolas
Identify 𝑎, 𝑏, and the transverse axis. Then graph.
1) 𝑦2
4−
𝑥2
16= 1
Graph Hyperbolas
Identify 𝑎, 𝑏, and the transverse axis. Then graph.
2) 𝑥2
16−
𝑦2
9= 1
Graph Hyperbolas
Identify 𝑎, 𝑏, and the transverse axis. Then graph.
3) 𝑥2 −𝑦2
25= 1
Graph Hyperbolas
To find the vertices of hyperbolas
1. Identify the center, 𝑎, and the transverse axis
2. ±𝑎 to the center along the transverse axis
Graph Hyperbolas
Find the center, 𝑎, and both vertices (𝑉1, 𝑉2)
4) 𝑦2
4−
𝑥2
16= 1 5)
𝑥2
16−
𝑦2
9= 1
6) 𝑥2 −𝑦2
25= 1
Graph Hyperbolas
The focal points (foci) are a set distance 𝒄 away
from the center along the transverse axis.
𝑐2 = 𝑎2 + 𝑏2
Graph Hyperbolas
Find the center, 𝑐, and both foci (𝐹1, 𝐹2)
7) 𝑦2
4−
𝑥2
16= 1 8)
𝑥2
16−
𝑦2
9= 1
Equation of Hyperbolas
Find the equation of the hyperbola with the
following information.
9) Vertices (±3,0)
Foci (±5,0)
Equation of Hyperbolas
Find the equation of the hyperbola with the
following information.
10) Vertices (0, ±12)
Foci (0, ±13)