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10.4(a) Notes: The Hyperbola
Lesson Objective: Graph and write the equations of hyperbolas in standard form.
CCSS: G.GPE.3
You will need: colored pens, ruler
Real-World App: Where is an explosion located?
This is Jeopardy!!!: These are the foci of an
ellipse whose equation is: 4x2 + 9y2 – 32x + 36y
+ 64 = 0
Lesson 1: The Anatomy of a Hyperbola
Hyperbola: The set of all points, P, in a
plane the difference of whose distances
from 2 foci, F1 and F2, is constant.
Graph the hyperbola whose equation is:
(x – 0)2 – (y – 0)2 = 1
16
9
Graph asymptotes y = + 3/4 x.
Plot foci F1 at (-5, 0) and F2 at (5, 0).
Lesson 1: The Anatomy of a Hyperbola
(x – 0)2 – (y – 0)2 = 1
16
9
Graph asymptotes y = + 3/4 x.
Plot foci F1 at (-5, 0) and F2 at (5, 0).
Lesson 1: The Anatomy of a Hyperbola
Hyperbola Equation in Standard Form:
(x – h)2 – (y – k)2 = 1 Asymptotes:
a2
b2
y – k = + b/a(x – h)
Lesson 1: The Anatomy of a Hyperbola
(y – k) 2 – (x – h)2 = 1
a2
b2
Asymptotes:
y – k = + a/b(x – h)
Lesson 1: The Anatomy of a Hyperbola
Hyperbola Equation in Standard Form:
(x – h)2 – (y – k)2 = 1
(y – k) 2 – (x – h)2 = 1
a2
b2
a2
b2
Lesson 1: The Anatomy of a Hyperbola
Hyperbola Equation in Standard Form:
(x – h)2 – (y – k)2 = 1
(y – k) 2 – (x – h)2 = 1
a2
b2
a2
b2
Center: (h, k); midpoint of transverse axis
Asymptotes: lines that pass through center and
which the graph follows
Lesson 1: The Anatomy of a Hyperbola
Hyperbola Equation in Standard Form:
(x – h)2 – (y – k)2 = 1
(y – k) 2 – (x – h)2 = 1
a2
b2
a2
b2
Transverse Axis: The axis on which the vertices,
a, and foci, c, lie; length = 2a
Vertices: +a units from the center on the
transverse
Foci: +c units from the center on the transverse
axis; c2 = a2 + b2
Lesson 1: The Anatomy of a Hyperbola
Hyperbola Equation in Standard Form:
(x – h)2 – (y – k)2 = 1 (y – k)2 – (x – h)2 = 1
a2
b2
a2
b2
Center
Asymptotes
Transverse
Axis
Vertices
Foci
(h, k)
(h, k)
y – k = +b/a(x – h) y – k = + a/b(x – h)
Parallel to x-axis,
a is under (x – h)2
Parallel to y-axis,
a is under (y – k)2
(h + a, k)
(h + c, k)
c2 = a2 + b2
(h, k + a)
(h, k + c)
c2 = a2 + b2
Lesson 2: Graphing Hyperbolas at (0, 0)
Graph and find the vertices and the foci:
4y2 – x2 = 4
Lesson 3: Graphing Hyperbolas at (h, k)
Graph and find the asymptotes and foci:
4x2 – 9y2 – 24x – 90y – 153 = 0
10.4(a): Do I Get It? Yes or No
Graph. Find the asymptotes, vertices and foci.
1. x2 – y2 = 1
25 16
2. 9y2 – 4x2 = 36
3. 4x2 – 25y2 – 24x + 250y – 489 = 0