Multiplying and Dividing Rational Expressions
What is a restriction and how do you solve for one
A value that the denominator cannot equal so that the equation/expression is not undefined. Solved by equating each expression in the denominator to 0.
What must you do before adding for subtracting fractions?
Common denominator
Define a rational equations
An equation containing at least one fraction where the numerator, denominator, or both are polynomials.
How do you find the x and y intercepts in a rational function?
x-intercept: set the numerator to zero
y-intercept: plug in 0 for x and solve
How does parts of a rational function affect key features?
Domain: affected by x of hole and VA
Range: affected by y of hole and HA
End Behavior: affected by HA
Simplify the following:
(2/3)*(9/14)
3/7
Method: Cross multiplication
Simplify: ((2x)/(x-1))+((3)/(x+5)) AND state restrictions
(2x2+13x-3)/((x-1)(x+5))
Restrictions: x cannot equal 1,-5
Solve for x
(x/3)-(1/12)=5/6
x=4/11
How do you solve for the Vertical Asymptote?
Factor the denominator and set each factor equal to 0.
Given a graph, what are the steps to writing an equation
1. Write factors from zeros in numerator
2. Write factors from vertical asymptote in denominator
3. Include hole factors in both numerator & denominator
4. Use given point to solve for "a"
Simplify: ((4x+3)/(x-5))*((x-5)/(x+3)) AND state restrictions
(4x+3)/(x+3)
Restrictions: x is not equal to 5,-3
((2x-1)/(3x2+13x+4))+((x+3)/(x2-3x-28)) AND state restrictrions
(5x2-5x+10)/((3x+1)(x+4)(x-7))
Restrictions: x cannot be -1/3, -4, 7
Solve for x and state restrictions:
(4x+1)/(x+1)=(12/(x2-1))+3
x=5,-2
Restrictions: x cannot equal 0, -1
What is the difference between Horizontal Asymptote and Slant Asymptote?
IF: deg. denominator>deg. of numerator BOTTOM HEAVY y=0 HA
IF: deg. denominator= deg. denominator y=-1 HA
IF: deg. denominator< deg. numerator TOP HEAVY, do long division and ignore remainder
Write the equation of the functions given the following key features. Sketch it on your own paper:
- VA is x=2 and x=1
- no x intercept
- y intercept is (0,-2)
-4/((x-2)(x-1))
Simplify: ((x+2)/(9x-1))/((2x+1)/(9x-1)) AND state restrictions
(x+2)/(2x+2)
Restrictions: x is not equal to 1/9, -1
(2/(x+4))-((4x-x2)/(x2-16)) AND state restrictions
(2+x)/(x+4)
Restrictions: x cannot equal -4,4
Solve for x and find restrictions
((x-1)/(x-2))+((3x+6)/(2x+1))=3
x=7,1
Restrictions: x cannot equal 2,-1/2
What is a hole?
Exist when the function has the same factor on both the top and bottom.
Set equal to 0 to find the hole, graph it with open circle and then cross it out on top and bottom to simplify the equation.
Write the equation of the functions given the following key features. Sketch it on your own paper:
- x int: (-1,0) and (-3,0)
- VA is x=0
- passes through the point (2,22.5)
(3(x+1)(x+3))/x
Simplify:
((2x2+x-10)/(x2+2x-8))/(4x2+20x+25)/(x+4))
AND state restrictions
1/(2x+5)
Restrictions: x is not equal to 2, -4, -5/2
(2/(x+2))/((1/(x+2))+(2/x)) AND state restrictions
(2x)/(3x+4)
Restrictions: x cannot equal -2, 0, -4/3
Solve for x and find restrictions
(2/(x+3))-((3/(4-x))=((2x-2)/(x2-x-12))
x=-1
Restrictions: x cannot equal -3,4
What are the steps to graphing a rational function?
1. Factor the rational function (if possible)
2. Find the hole (if possible) and plot it on the graph with an open dot
3. Find and plot the x and y intercepts
4. Find and draw the Vertical Asymptote using dotted lines
5. Find and draw the Horizontal OR Slant Aysmptote using a dotted line.
6. Find and plot enough points to have at least 3 points on either side of your VA(s)
7. Draw the function
Write the equation of the functions given the following key features. Sketch it on your own paper:
- range: (-inf, 2)U(2,4)U(4,inf)
- end behavior: x-->-inf, y-->4 and x-->inf, y-->4
- VA is x=-5
(4x(x-5))/((x+5)(x-5))