# Graph (y^2-2y+1)/(2y-3) y2-2y+12y-3
Find where the expression (y-1)22y-3 is undefined.
y=32
x=(y-1)22y-3 is an equation of a line, which means there are no horizontal asymptotes.
No Horizontal Asymptotes
Find the oblique asymptote using polynomial division.
Factor using the perfect square rule.
Rewrite 1 as 12.
y2-2y+122y-3
Check the middle term by multiplying 2ab and compare this result with the middle term in the original expression.
2ab=2⋅y⋅-1
Simplify.
2ab=-2y
Factor using the perfect square trinomial rule a2-2ab+b2=(a-b)2, where a=y and b=-1.
(y-1)22y-3
(y-1)22y-3
Expand (y-1)2.
Rewrite (y-1)2 as (y-1)(y-1).
(y-1)(y-1)2y-3
Apply the distributive property.
y(y-1)-1(y-1)2y-3
Apply the distributive property.
y⋅y+y⋅-1-1(y-1)2y-3
Apply the distributive property.
y⋅y+y⋅-1-1y-1⋅-12y-3
Reorder y and -1.
y⋅y-1y-1y-1⋅-12y-3
Raise y to the power of 1.
y⋅y-1y-1y-1⋅-12y-3
Raise y to the power of 1.
y⋅y-1y-1y-1⋅-12y-3
Use the power rule aman=am+n to combine exponents.
y1+1-1y-1y-1⋅-12y-3
Add 1 and 1.
y2-1y-1y-1⋅-12y-3
Multiply -1 by -1.
y2-1y-1y+12y-3
Subtract 1y from -1y.
y2-2y+12y-3
y2-2y+12y-3
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of 0.
 2y – 3 y2 – 2y + 1
Divide the highest order term in the dividend y2 by the highest order term in divisor 2y.
 y2 2y – 3 y2 – 2y + 1
Multiply the new quotient term by the divisor.
 y2 2y – 3 y2 – 2y + 1 + y2 – 3y2
The expression needs to be subtracted from the dividend, so change all the signs in y2-3y2
 y2 2y – 3 y2 – 2y + 1 – y2 + 3y2
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 y2 2y – 3 y2 – 2y + 1 – y2 + 3y2 – y2
Pull the next terms from the original dividend down into the current dividend.
 y2 2y – 3 y2 – 2y + 1 – y2 + 3y2 – y2 + 1
Divide the highest order term in the dividend -y2 by the highest order term in divisor 2y.
 y2 – 14 2y – 3 y2 – 2y + 1 – y2 + 3y2 – y2 + 1
Multiply the new quotient term by the divisor.
 y2 – 14 2y – 3 y2 – 2y + 1 – y2 + 3y2 – y2 + 1 – y2 + 34
The expression needs to be subtracted from the dividend, so change all the signs in -y2+34
 y2 – 14 2y – 3 y2 – 2y + 1 – y2 + 3y2 – y2 + 1 + y2 – 34
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 y2 – 14 2y – 3 y2 – 2y + 1 – y2 + 3y2 – y2 + 1 + y2 – 34 + 14
The final answer is the quotient plus the remainder over the divisor.
y2-14+14(2y-3)
The oblique asymptote is the polynomial portion of the long division result.
y=y2-14
y=y2-14
This is the set of all asymptotes.
Vertical Asymptotes: y=32
No Horizontal Asymptotes
Oblique Asymptotes: y=y2-14
Graph (y^2-2y+1)/(2y-3)   ## Download our App from the store

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