Show:
Vertical Asymptotes
Horizontal Asymptotes
Holes
Zeros
Mixed
100
What is a vertical asymptote and what is an exception to having a vertical asymptote?
A vertical asymptote is when the denominator equals zero or x=something. If there is a hole at a certain point then the asymptote does not exist.
100
What is a horizontal asymptote and what are three ways using the degrees you can tell what the horizontal asymptote is?
- When x approaches infinity - y= something Look at the degrees: Numerator exponent is bigger= then it is none Numerator exponent is smaller=then it is 0 Exponents are the same= divide coefficients
100
What is a hole?
- It is where nothing can happen. - When the numerator and the denominator have a common factor it crosses out creating a hole
100
What are zeros?
-when the numerator equals zero -but make sure there are no holes like y= (x+5)(x+3)/(x+5)(x+7) There zero is only -3 because there is a hole at -5
100
Find the VA, HA, Hole, and Zero. f(x)=x^2+2x/-4x+8
x(x+2)/-4(x-2) VA= 2 HA= None Hole= None Zero= x=-2, x=0
200
Find the vertical asymptote. y= x^2+2x-3/x^2-5x-6
(x+3)(x-1)/(x+1)(x-5) VA= x=-1 and x=5
200
Find the horizontal asymptote. y= x+4/x^2+6x+8
HA= 0
200
Find the hole. y= x^2-x-2/x-2
(x-2)(x+1)/(x-2) Hole= x=2
200
Find the zero. y= x+4/x^2+6x+8
(x+4)/(x+4)(x+2) There are no zeros because x+4 is common among the numerator and denominator therefore there is a hole there. NONE!
200
Find the VA, HA, Hole, and Zero. f(x)=x^3-16x/-4x^2+4x+24
VA= x=3, x=-2 HA= None Hole= None Zero= x=-4, x=4, x=0
300
Find the vertical asymptote. y= x^2+2x-3/x^2-5x-6
(x+3)(x-1)/(x+1)(x-5) VA= x=-1 and x=5
300
Find the horizontal asymptote. y=3x^3+10/2x^2-16
HA= None
300
Find the hole. y=x+4/x^2+6x+8
(x+4)/(x+4)(x+2) Hole= x=-4
300
Find the zero. y= x^2-3x-10/x^2+x-2
(x-5)(x+2)/(x+2)(x-1) Zero= 5 only There is a hole at -2 so there can't be a zero there.
300
Find the hole. y= 9/15x-15
There are no holes because the numerator will never equal zero. For there to be a hole the numerator and denominator must equal zero.
400
Find the vertical asymptote. y= x^3-8/x^2+5x+6
VA= x=-2, x=-3
400
Find the horizontal asymptote. y=3x^2+x-6/x^2+8x+15
HA= 3
400
Find the hole. y=x^2-3x-10/x^2+x-2
(x+2)(x-5)/(x+2)(x-1) Hole= -2 because the numerator and denominator have x+2 in common
400
Find the zero. y= x^2+15x+56/x^2-2x-35
(x+8)(x+7)/(x+5)(x-7) Zero= x=-8, x=-7
400
Graph the following. y=1/x^2
Graph on the board.
500
Find the vertical asymptote. y= x^3-4x^2+4x/x^2-16
x(x^2-4x+4)/(x+4)(x-4) x(x-2)(x-2)/(x+4)(x-4) VA= x=0, x=2
500
Find the horizontal asymptote. y= 3x^4-5x^2+4/2x^4-2x^3+5
HA= 3/2
500
Find the hole. f(x)=3x^2-12x/x^2-2x-3
3x(x-4)/(x+1)(x-3) None
500
Find the zero. y= x^2+3x+2/x^2+5x+6
(x+2)(x+1)/(x+3)(x+2) Zero=-1 only because -2 is a hole
500
Graph the following. y= -1/x
Graph on board.