293031RationalFunctions Flashcards
Solve:
-3/x² * -5/12x7
-3/x² * -5/12x7
(-3)*(-5)/(x²*12x7)
= 15/12x9
= 5/4x9
Solve:
(3x + 1)/4 * 8x/(9x² - 1)
(3x + 1)/4 * 8x/(3x-1)(3x + 1)
1/1 * 2x/(3x-1)
2x/(3x-1)
Solve:
(3x + 7)/(5x + 10) * (x² + 7x + 10)
(3x + 7)/(5x + 10) * (x² + 7x + 10)
(3x + 7)/5(x + 2) * (x² + 5)(x + 2)
(3x + 7)(x² + 5)/5
What is the formula for dividing rational expressions?
a/b ÷ c/d
= a/b * d/c
Flip one of the expressions.
Solve:
(x² + 13x + 40)/(x - 7) ÷ (x + 8)/(x² - 49)
(x² + 13x + 40)/(x - 7) ÷ (x + 8)/(x² - 49)
(x² + 13x + 40)/(x - 7) * (x² - 49)/(x + 8)
(x + 8)(x + 5)/(x - 7) * (x + 7)(x - 7)/(x + 8)
(x + 5)(x + 7)
Both denominators were elimiated.
Solve:
(x² + 10x - 11)/(x² + 12x + 11) ÷ (x - 1)
(x² + 10x - 11)/(x² + 12x + 11) * 1/(x - 1)
(x + 11)(x - 1))/(x + 11)(x + 1)) * 1/(x - 1)
(x - 1)/(x + 1) * 1/(x - 1)
= 1/(x + 1)
Solve:
(2x + 3)/(x - 1) - (x + 2)/(x + 1)
(2x + 3)/(x - 1) - (x + 2)/(x + 1)
Find common denominator:
(2x + 3)(x + 1) - [(x + 2)((x - 1)/[(x - 1)(x + 1)]
2x² + 5x + 3 - [x² + x - 2]/[(x - 1)(x + 1)]
(x² + 4x + 5)/[(x - 1)(x + 1)]
When graphing rational expressions, how do you find the roots (x-intercepts)?
Find the roots of the top polynomial when the equations has been simplified to its lowest terms.
When does y = 0 (x-intercept)?
When the numerator = 0.
What are the roots (x-intecepts) of:
y = (x² - 11x + 24)/(x² + x - 20)?
y = (x² - 11x + 24)/(x² + x - 20)?
y = (x - 8)(x - 3) ÷ (x + 5)(x - 4)
It can’t be simplified.
x-intercepts = (8,0) and (3,0)
What are the roots (x-intercepts) of:
y = (x - 2)(x + 3))/(x - 5)(x + 1)?
It can’t be simplified.
roots = (2,0) and (-3,0)
What are the roots of:
1/(x - 2)?
It has no roots.
What are the roots (x-intercepts) of:
y = (x - 5)(x + 3))/(x - 5)(x + 7)?
y = (x - 5)(x + 3))/(x - 5)(x + 7)
Root = (-3,0)
What are the vertical asymptotes of:
1/(x - 2)?
Vertical asymptotes are the numbers that x can’t be.
x = 2
How do you find a vertical asymptote?
(1) Cancel (simplify) everything you can.
(2) Find the values that make the denominator 0.
What are the Vertical Asymptotes of:
y = (x² - 4) / (x² - 9)?
What are its x-intercepts?
y = (x² - 4) / (x² - 9)
y = (x - 2)(x + 2) ÷ (x - 3)(x = 3)
It is simplified.
Vertical asymptotes = x = 3 and x = -3
x-intercepts = (2,0) and (-2,0)
where the numerator = 0
How do you find Horizontal Asymptotes?
(1) “Erase” all terms in the numerator and denominator which are not “dominate terms”._
(Dominant terms are the one with the highest power)
If the dominant terms are the same, the horizontal asymptote is the ratio of the coefficients.
If the dominant term in the numerator is larger than the one in the denominator, there is no horizontal asymptote.
If the dominant term in the numerator is smaller than the one in the denominator, the horizontal asymptote is 0`.
y = 4/(x + 3)
What are the x-intercepts?
Vertical asymptote?
Horizontal asymptote?
y-intercept
There are no x-intercepts. (Numerator just 4)
Vertical asymptote at x = -3
Horizontal asymptote = 0. x0/x1
y-intercept (x = 0) = 4/3
y = (5x - 15)/(x + 2)
What are the x-intercepts?
Vertical asymptote?
Horizontal asymptote?
y-intercept
y = (5x - 15)/(x + 2)
=5(x - 3)/(x + 2)
x-intercept (y = 0) at x = 3
Vertical asymptote (den = 0) = x = -2
Horizontal asymptote = 5
y-intercept (x = 0) = -15/2
Summariza the definitions of:
x-intercept
y-intercept
vertical asymptote
horizontal asymptote
x-intercept: roots of the numerator
y-intercept: x = 0
vertical asymptote: denominator = 0
horizontal asymptote: ratio of dominant terms
