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Warm Up #5
CHAPTER 9:RATIONAL FUNCTIONS
9.1 INVERSE VARIATION
Review of Direct Variation
Direct Variation:
A function of the form y = kx such that as x increases y increases. And k is the constant of variation.
EX: If x and y vary directly, and x = 6 when y = 3, write an equation.
Inverse Variation
Inverse Variation:
For inverse variation a function has the form
Where k is a constant
As one value of x and y increase, the other decreases!
Modeling Inverse Variation
Suppose that x and y vary inversely, and x = 3 when y = -5. Write the function that models the inverse variation.
Modeling Inverse Variation
Suppose that x and y vary inversely, and x = -2 when y = -3. Write the function that models the inverse variation.
Rational Functions
Identifying from a table
Example
Combined Variation
A combined Variation has more than one relationship.
EX: is read as y varies directly with
x (on top) and inversely with z (on the bottom).
Examples
Write the function that models each relationship.
1. Z varies jointly with x and y. (Hint jointly means directly)
2. Z varies directly with x and inversely with the cube of y
3. Z varies directly with x squared and inversely with y
Write a function
Z varies inversely with x and y.
Write a function when x = 2 and y = 4 and z = 2
Warm Up #6
HW Check – 9.1 #22-32
9.2 AND 9.3 GRAPHING, ASYMPTOTES
Investigation
Graph the following:
1. Y = 3/X 2. Y = 6/X
3. Y = -8/X 4. Y = -4/X
What do you notice?!?!?
Rational Function
Rational functions in the form y = k/x is split into two parts. Each part is called a BRANCH.
If k is POSITIVE the branches are in Quadrants I and III
If k is NEGATIVE the branches are in Quadrant II and IV
Asymptotes - An Asymptote is a line that the graph approaches but NEVER touches.
Horizontal Asymptote
Vertical Asymptote
Asymptotes
From the form
The Vertical Asymptote is x = b
The Horizontal Asymptote is y = c
Identifying Asymptotes
Identify the Asymptote from the following functions.
1. 2.
3. 4.
Translating
Translate y=3/x
1. Up 3 units and Left 2 Units
2. Down 5 units and Right 1 unit
3. Right 4 units
4. Such that it as a Vertical asymptotes of x=3 and a horizontal asymptote of y= -2
Rational Functions
A rational function can also be written in the form
where p and q are polynomials.
Asymptotes
Vertical Asymptotes are always found in the BOTTOM of a rational function.
Set the bottom equal to zero and solve! This is a Vertical Asymptote!!
Asymptotes
Find the Vertical Asymptotes for the following.
1. 2.
3. 4.
Asymptotes
What is the Asymptotes?
Graph it, what do you notice?!
Holes
A HOLE in the graph is when (x – a) is a factor in both the numerator and the denominator.
So on the graph, there is a HOLE at 4.
Continuous and Discontinuous
A graph is Continuous if it does not have jumps, breaks or holes.
A graph is Discontinuous if it does have holes , jumps or breaks.
Discontinuous
Find the places of Discontinuity!
Discontinuous
Find the places of Discontinuity!
Discontinuous
Find the places of Discontinuity!
Discontinuous
Find the places of Discontinuity!
9.4Simplifying, Multiplying and Dividing
Simplest Form
A rational expression is in SIMPLEST FORM when its numerator and denominator are polynomials that have no common divisors.
When simplifying we still need to remember HOLES as points of discontinuity.
Examples
Simplify
1.
Examples
Simplify
1.
Examples
Simplify
1.
Examples
Simplify
1.
Multiplying Rational Expressions
Multiply the tops and the bottoms. Then Simplify.
Ex:
Multiplying Rational Expressions
Multiply
Dividing Rational Expressions
Remember: Dividing by a fraction is the same thing as multiplying by the reciprocal. Flip the second fraction then multiply the tops and the bottoms. Then Simplify.
Ex:
Dividing Rational Expressions
Divide
9.5 ADDING AND SUBTRACTING
Review of Multiplying
Multiply the following:
1. 2.
2. .
Review of Dividing
Divide the following:
1. .
2. .
Review of Adding and Subtracting Fractions With Common Denominators
Adding
.
Subtracting
.
9.5 ADDING AND SUBTRACTING WITH UNLIKE DENOMINATORS
Inverse Variation
1. Given that x and y vary inversely, write an equation for when y = 3 and x = -4
2. Write a combination variation equation for: z varies jointly with x and y, and inversely with w.
3. Write a combination variation equation for: y varies directly with x and inversely with z.
Common Denominator
A Common Denominator is also the LEAST COMMON MULTIPLE.
LCM is the smallest numbers that each factor can be divided into evenly.
Finding LCM of 2 numbers:
7:
21:
LCM
Find the LCM of each pair of numbers
1. 4, 5
2. 3, 8
3. 4, 12
Review of Adding and Subtracting Fractions With Unlike Denominators
Add or subtract the following.
1.
2.
Steps for Adding or Subtracting
Finding LCM
For Variables LCM: Take the Largest Exponent!
Find the LCM:
x4 and x
x3 and x2
3x5 and 9x8
Adding and Subtracting Fractions With Unlike Denominators
Add or subtract the following.
1.
2.
Finding LCM of Expressions
For Expressions LCM: Include ALL factors!!
EX:• 3(x + 2) and 5(x – 2)
• x(x + 4) and 3x2(x + 4)
• (x2 + 2x - 8) and (x2 – 4)
• 7(x2 – 25) and 2(x2 + 7x + 10)
Add or Subtract the Following
1.
Add or Subtract the Following
1.
Add or Subtract the Following
1.
Add or Subtract the Following
1.
Complex Fractions
A complex fraction is a faction that has a fraction in its numerator or denominator.
Simplifying
To Simplify Multiply the top and the bottom by the COMMON DENOMINATOR
Try Some!
Simplify
9.5 SOLVING RATIONAL EQUATIONS
Proportions
Remember to solve proportions you cross multiply!
Example
Solving
Use cross multiplication to solve. Check for Extraneous Solutions when variables are in the bottom because we cannot divide by zero!
Try Some!
Solve the following.
Try Some!
Solve the following.
Try Some!
Solve the following.
Sum or Difference Equations
Solve the equation for x.
Eliminate the Fraction
We can eliminate the fractions all together if we multiply the whole equation by the LCM of the denominators!
Examples
Solve.
Examples
Solve.
