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Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Jan 02, 2016

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Page 1: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Warm Up #5

Page 2: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

CHAPTER 9:RATIONAL FUNCTIONS

Page 3: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

9.1 INVERSE VARIATION

Page 4: 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.

Page 5: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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!

Page 6: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Modeling Inverse Variation

Suppose that x and y vary inversely, and x = 3 when y = -5. Write the function that models the inverse variation.

Page 7: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 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.

Page 8: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Rational Functions

Page 9: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Identifying from a table

Page 10: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Example

Page 11: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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).

Page 12: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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

Page 13: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Write a function

Z varies inversely with x and y.

Write a function when x = 2 and y = 4 and z = 2

Page 14: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Warm Up #6

Page 15: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

HW Check – 9.1 #22-32

Page 16: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

9.2 AND 9.3 GRAPHING, ASYMPTOTES

Page 17: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Investigation

Graph the following:

1. Y = 3/X 2. Y = 6/X

3. Y = -8/X 4. Y = -4/X

What do you notice?!?!?

Page 18: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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

Page 19: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Asymptotes - An Asymptote is a line that the graph approaches but NEVER touches.

Horizontal Asymptote

Page 20: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Vertical Asymptote

Page 21: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Asymptotes

From the form

The Vertical Asymptote is x = b

The Horizontal Asymptote is y = c

Page 22: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Identifying Asymptotes

Identify the Asymptote from the following functions.

1. 2.

3. 4.

Page 23: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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

Page 24: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Rational Functions

A rational function can also be written in the form

where p and q are polynomials.

Page 25: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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!!

Page 26: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Asymptotes

Find the Vertical Asymptotes for the following.

1. 2.

3. 4.

Page 27: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Asymptotes

What is the Asymptotes?

Graph it, what do you notice?!

Page 28: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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.

Page 29: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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.

Page 30: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Discontinuous

Find the places of Discontinuity!

Page 31: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Discontinuous

Find the places of Discontinuity!

Page 32: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Discontinuous

Find the places of Discontinuity!

Page 33: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Discontinuous

Find the places of Discontinuity!

Page 34: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

9.4Simplifying, Multiplying and Dividing

Page 35: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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.

Page 36: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Examples

Simplify

1.

Page 37: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Examples

Simplify

1.

Page 38: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Examples

Simplify

1.

Page 39: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Examples

Simplify

1.

Page 40: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Multiplying Rational Expressions

Multiply the tops and the bottoms. Then Simplify.

Ex:

Page 41: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Multiplying Rational Expressions

Multiply

Page 42: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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:

Page 43: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Dividing Rational Expressions

Divide

Page 44: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

9.5 ADDING AND SUBTRACTING

Page 45: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Review of Multiplying

Multiply the following:

1. 2.

2. .

Page 46: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Review of Dividing

Divide the following:

1. .

2. .

Page 47: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Review of Adding and Subtracting Fractions With Common Denominators

Page 48: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Adding

.

Page 49: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Subtracting

.

Page 50: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

9.5 ADDING AND SUBTRACTING WITH UNLIKE DENOMINATORS

Page 51: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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.

Page 52: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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:

Page 53: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

LCM

Find the LCM of each pair of numbers

1. 4, 5

2. 3, 8

3. 4, 12

Page 54: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Review of Adding and Subtracting Fractions With Unlike Denominators

Add or subtract the following.

1.

2.

Page 55: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Steps for Adding or Subtracting

Page 56: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Finding LCM

For Variables LCM: Take the Largest Exponent!

Find the LCM:

x4 and x

x3 and x2

3x5 and 9x8

Page 57: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Adding and Subtracting Fractions With Unlike Denominators

Add or subtract the following.

1.

2.

Page 58: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

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)

Page 59: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Add or Subtract the Following

1.

Page 60: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Add or Subtract the Following

1.

Page 61: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Add or Subtract the Following

1.

Page 62: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Add or Subtract the Following

1.

Page 63: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Complex Fractions

A complex fraction is a faction that has a fraction in its numerator or denominator.

Page 64: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Simplifying

To Simplify Multiply the top and the bottom by the COMMON DENOMINATOR

Page 65: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Try Some!

Simplify

Page 66: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

9.5 SOLVING RATIONAL EQUATIONS

Page 67: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Proportions

Remember to solve proportions you cross multiply!

Example

Page 68: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Solving

Use cross multiplication to solve. Check for Extraneous Solutions when variables are in the bottom because we cannot divide by zero!

Page 69: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Try Some!

Solve the following.

Page 70: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Try Some!

Solve the following.

Page 71: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Try Some!

Solve the following.

Page 72: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Sum or Difference Equations

Solve the equation for x.

Page 73: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Eliminate the Fraction

We can eliminate the fractions all together if we multiply the whole equation by the LCM of the denominators!

Page 74: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Examples

Solve.

Page 75: Warm Up #5. CHAPTER 9: RATIONAL FUNCTIONS 9.1 INVERSE VARIATION.

Examples

Solve.