Intermediate Algebra (11th edition)

652 CHAPTER 11 Nonlinear Functions, Conic Sections, and Nonlinear Systems

NOTEAs with circles and ellipses, hyperbolas are graphed with a graphing calculator

by first writing the equations of two functions whose union is equivalent to the equa-

tion of the hyperbola. A square window gives a truer shape for hyperbolas, too.

SUMMARY OF CONIC SECTIONS

Equation Graph Description Identification

or

or

y 2

b 2

x^

2

a 2

 1

x 2

a 2



y 2

b 2

 1

x 2

a 2



y 2

b 2

 1

1 yk 22 r 2

1 xh 22 

xa 1 yk 22 h

xay 2 byc

ya 1 xh 22 k

yax^2 bxc It opens up if

down

if

The vertex is

It opens to the

right if

to the left if

The vertex is

The center is

and the radius is r.

The x-intercepts

are and

.

The y-intercepts

are and

The x-intercepts

are and

.

The asymptotes

are found from

and

The y-intercepts

are and

The asymptotes

are found from

and

1 - a, b 2.

1 - a, -b 2 ,

1 a, b 2 , 1 a, -b 2 ,

1 0, -b 2.

1 0, b 2

1 - a, b 2.

1 - a, -b 2 ,

1 a, b 2 , 1 a, -b 2 ,

1 - a, 0 2

1 a, 0 2

1 0, -b).

1 0, b 2

1 - a, 0 2

1 a, 0 2

1 h, k 2 ,

1 h, k 2.

a 6 0.

a 7 0,

1 h, k 2.

a 6 0.

a 7 0,

y

(h, k) x

a > 0

0

Parabola

y

x

(h, k)

a > 0

0

Parabola

y

x

(h, k)

(^0) r
Circle
y
(–a, 0) 0 x
(0, –b)
(0, b)
(a, 0)
Ellipse
y
0 x
(–a, 0) (a, 0)
Hyperbola
y
0 x
(0, –b)
(0, b)
Hyperbola
It has an -term.
yis not squared.
It has a -term.
xis not squared.

• and -terms
  have the
  same positive
  coefficient.


• and -terms
  have different
  positive
  coefficients.

has a positive

coefficient.

has a negative

coefficient.

has a positive

coefficient.

has a negative

coefficient.

x 2

y 2

y 2

x 2

x 2 y 2

x 2 y 2

y 2

x^2