Symmetry and Asymptotes

f(-x) = f(x) Even Symmetrical wrt y-axisf(-x) = -f(x) Odd Symmetrical wrt origin

2y x 2 2xy x Even

1yx 1

y1x

1

Neither

3

1yx x

3x

1yx

3

1x x

Odd

y cos x y s xco cos x Even

y sinx y n xsi sinx Odd

Asymptotes

VERTICAL – values of x which make the denominator 0

HORIZONTAL – take the limit as x approaches infinity of f(x)

OBLIQUE OR SLANT – will occur when degree of numerator is ONE MORE than degree of the denominator

2x 1y2x 4

1 x 22

Slant asymptote is 1y x 22

3

2

x 4yx 1

y x Slant asymptote is y x

x 3yx 2

a. Find the symmetry

3y xx 2

None

b. Find the intercepts 3 x 3y y 1.50 00

x 32 x 2

c. Find the asymptotes

No slant x 2 0 x 2 x

x 3lim 1 y 1x 2

d. Find the ‘number line’

2 2

1 x 2 1 x 3 1y 'x 2 x 2

3

3

2y " 2 x 2x 2

-2

y’y’’ +

_ __

x 3yx 2

No Symmetry

Asymptotes x = -2 y = 1

-2y’y’’ +

_ __

(0, 1.5) (-3, 0)

2x x 2yx 2

a. Find the symmetry 2x x 2

y2x

None

b. Find the intercepts x 2 x 12y y 10 0 0 x 2,1

2 x 20

c. Find the asymptotes

x 2 0 x 2 2

x

x x 2limx 2

y x 3

d. Find the ‘number line’

2 2

2 2 2

2x 1 x 2 1 x x 2 x x 4x 4xy 'x 2 x 2 x 2

2 2

4

2

3 3

2x 4 x 2 2 x 2 x 4xy "

x 2

2x 4 x 2 2 x 4x 8

x 2 x 2

+_ __

0y’y’’ 2 4

+ _ + +

2x x 2yx 2

Symmetry – None (0, 1) (-2, 0) (1, 0)Asymptotes y = x + 3 x = 2

+_ __

0y’y’’ 2 4

+ _ + +