Algebra 1 Unit 13

Chapter 12.1-12.9

Guided Notes

NAME __________________________ Period _____________

Teacher __________________

Date: ________________ 12.1 Inverse Variation

Notes Direct Variation: When y __________as x ___________. Written as ___________ where k is a nonzero constant. Inverse Variation: y varies inversely as x (y __________ as x ___________ or vice versa) if there is some nonzero constant k where it can be written _____________ Constant of variation = ______ Example 1: The owner of Superfast Computer Company has calculated that the time t in hours that it takes to build a particular model of computer varies inversely with the number of people p working on the computer. The equation pt = 12 can be used to represent the people building a computer. Complete a table and draw a graph of the relation. p 2 4 6 8 10 12 t

p

t

Example #2 Graph an inverse variation in which y varies inversely as x and y = 15 when x = 6 Use inverse variation equation: xy=k Substitute and solve for k So equation becomes xy= ___ Find values to graph:

x y -9 -6 -3 -2 0 2 3 6 9

The graphs of a direct variation equations are in the form y=kx and are linear. The graphs of inverse variation equations are not linear in the form 8*3 and are curves that approach a value. In example #1, the p and t cannot be zero so the graph will never touch the p or t axis. In example #2, neither x nor y can be 0 as it would be undefined. The graph approaches these axis lines.

Example #3 Graph an inverse variation in which y varies inversely as x and y = 1 when x = 4 Use inverse variation equation: xy=k Substitute and solve for k So equation becomes xy= ___ Find values to graph:

x y -4 -2 -1 0 1 2 4

If (x1, y1) and (x2, y2) are solutions of an inverse variation, then x1y1=k and x2y2=k. This means that ___________ = ___________. This is the product rule for inverse variations and can be used to form proportions to solve inverse variation problems. x1y1 = x2y2 product rule x1y1 = x2y2 divide each side by x2y1 x2y1 x2y1 x1 = y2 simplify x2 y1

Example #4 If y varies inversely as x and y = 4 when x = 7, find x when y = 14. Let x1 = 7, y1= 4, and y2 = 14. Solve for x2 x1y1 = x2y2 Example #5 If y varies inversely as x and y = 8 when x = 6, find y when x = 4. Let x1 = 6, y1= 8, and x2 = 4. Solve for y2 x1y1 = x2y2 Example #6 Anna is designing a mobile to suspend from a gallery ceiling in art class. A chain is attached eight inches from the end of a bar that is 20 inches long. On the shorter end of the bar is a sculpture weighing 36 kilograms. She plans to place another piece of artwork on the other end of the bar. How much should the second piece of art weigh if she wants the bar to be balanced?

Date: ________________

12.2 Rational Expressions Notes

Rational expressions are algebraic expressions that can be written as a __________. The numerator and denominator are polynomials. Some examples are: Denominators of a fraction can not be ________. Excluded Values – values of a variable that result in a ____________ of __________ and must be excluded from the ________ of the variable. Example #1 State the excluded value of each of the following: 3b-2 5 12x 5a2+2 b+7 a-4 3x-1 a2-a-12

We simplify rational expressions the same way we simplify fractions, eliminate any common factors of the numerator and denominator. Example #2 Simplify -7a2b3 21a5b -7a2b3 = (7a2b)(-b2) The GCF of the numerator and 21a5b (7a2b) (3a3) denominator is 7a2b = (-b2) Divide out the GCF (3a3) Example #3 Simplify 35xy2 14y2z Example #4 Simplify x2 – 2x – 15 x2 – x - 12

Example #5 Simplify 3x – 15 x2 – 7x + 10 Example #6 Simplify x2 – 9 x2 + 6x – 27

Date: ________________

12.3 Multiplying Rational Expressions Notes

Recall that to multiply rational numbers expressed as fractions, you multiply numerators and multiply denominators, then simplify the result. We do the same for multiplying rational expressions. Example #1 Find 5ab3 * 16c3 8c2 15a2b

5ab3 * 16c3 = 80ab3c3 8c2 15a2b 120a2bc2 = 40abc2(2b2c) the GCF of the numerator and 40abc2(3a) denominator is 40abc2. = 2b2c simplify 3a Example #2 Find 12xy2 * 27m3p 45mp2 40x3y

Example #3 Find x-5 * x2 x x2-2x-15 Example #4 Find a2 + 7a +10 * 3a + 3 a + 1 a + 2 Example #5 Find b+3 * b2 –4b + 3 4b -12 b2 – 7b - 30

Date: ________________ Section 12-4 – Dividing Rational Expression

Notes Recall that to divide rational numbers expressed as fractions you multiply by the reciprocal of the divisor. Keep Change Flip! Example #1 Find

5x2 ÷ 10x3 7 21

5x2 ÷ 10x3 = 5x2 * 21 7 21 7 10x3 = 5x2 * 21

7 10x3

= 3 2x

Example #2 Find n + 1 ÷ 2n + 2 n + 3 n + 4

3

2x

1

1

Example #3 Find 6x4 ÷ 24x 5 75 Example #4 Find 3m + 12 ÷ m + 4 m + 5 m – 2 Example #5 Find 12x - 36 ÷ (x - 3) x - 7

Example #6 Find w2 - 11 w – 26 ÷ w - 13 7 w + 7 Example #7 In 1986, Jenna Yeager and Dick Rutan piloted an experimental aircraft named Voyager around the world non-stop, without refueling. The trip took exactly 9 days and covered a distance of 24,012 miles. What was the speed of the aircraft in miles per hour?

Date: ________________ Section 12-5 – Dividing Polynomials

Notes DIVIDING POLYNOMIALS BY MONOMIALS: To divide a polynomial by a monomial, divide each ___________ of the polynomial by the monomial. Example #1 Find ( Example #2 Find Example #3 Find .

Example #4 Use Long Division to find ( Example #5 Use Long Division to find (

Date: ________________

Section 12-6 – Rational Expressions with Like Denominators Notes

Adding Rational Expressions: To add fraction with ____________ denominators, simply add the numerators, and write the sum over the common __________________. You can add rational expressions with like _________________ in the same way. Example #1 - Numbers in Denominator -

Find Example #2 – Binomials in Denominator -

Find Example #3 – Find an expression for the perimeter of rectangle PQRS.

P Q

S R

Subtract Rational Expressions: To subtract rational expressions with like ____________, subtract the numerator and write the difference over the ______________ denominator. Recall that to subtract an expression, you ________________ its additive ____________________. Example #4 – Subtract Rational Expressions –

Find Example #5 – Inverse Denominators -

Find

Date: ________________

Section 12-7 – Rational Expressions with Unlike Denominators Notes

Add Rational Expressions: The___________ _____________ ________________ (LCM) is the least number that is a common multiple of two or more numbers. Example #1 – LCM of Monomials – Find the LCM of and 18 . Example #2 – LCM of Polynomials - Find the LCM of Recall that to add fractions with _____________ denominators, you need to rename the fractions using the least common ________________ (LCM) of the denominators, known as the least common ______________(LCD). KEY CONCEPT – ADD RATIONAL EXPRESSIONS STEP 1 – STEP 2 – STEP 3 – STEP 4 – Example #3 – Monomial Denominators –

Find

Example #4 – Polynomial Denominators –

Find

Example #5 – Binomials in Denominators –

Find Example #6 – Polynomials in Denominators -

Find

Date: ________________

Section 12-8 – Mixed Expressions and Complex Fractions Notes

A number like is a ________________ number because it contains the sum of an

_________, 2, and a fraction, . An expression like 3 + is called a _______________ _________________ because it contains the sum of a monomial, 3, and a rational

expression Changing mixed expressions to rational expressions is similar to changing mixed number to improper fractions. Example #1 – Mixed Expression to Rational Expression –

Simplify 3 + KEY CONCEPT: Any Complex Fraction ---------, where b , can be expressed as -------. SIMPLIFY COMPLEX FRACTIONS: If a fraction has one or more fractions in the _______________ or _______________, it is called a _________________ _________________. You simplify an algebraic complex fraction the same way you simplify a ________________ complex fraction. Example #2 – Complex Fraction Involving Numbers – If Katelyn has pounds of cookie dough, and the average cookie requires of dough, how many cookies can she make?

Example #3 – Complex Fraction Involving Monomials - Simplify x2y2

a x2y

a2 Example #4 – Complex Fraction Involving Polynomials - Simplify a - 15 a – 2 a + 3 Example #5 – Complex Fraction Involving Polynomials - Simplify b - 2 b + 3 b – 4

Date: ________________

Section 12-9 – Solving Rational Equations Notes

______________ _______________- are equations that contain rational expressions. You can use ______________ products to solve rational equation, but only when both side so the equation are ___________ fractions. Example #1 – Use Cross Products -

Solve

Example #2 – Use the LCD – Solve n – 2 __ n – 3 = 1 n n – 6 n

Example #3 – Multiple Solutions - Solve - 4 + 3 = 1 a + 1 a Rational Equations can be used to solve __________ _____________. Example #4 – Work Problems Abbi has a lawn care service. One day, she asked her friend Jo-El to work with her. Normally, it takes Abbi two hours to mow and trim Mrs. Hart’s Lawn. When Jo-El works with her, the job took only 1 hour and 20 minutes. How long would it have taken Jo-El to do the job himself?

Rational equations can also be used to solve ______________ _______________. Example #5 – Rate Problem Two trains leave from different locations 9.46 miles apart and travel toward each other. The first train leaves at noon and arrives at its destination 24 minutes later. The second train leaves at noon at arrives at its destination 28 minutes later. At what time do the two trains pass each other? EXTRANEOUS SOLUTIONS: Multiplying each side of an equation by the LCD of two rational expressions can yield results that are not solutions to the original equation. Recall that such solutions are called ____________ __________________. Example #6 – No Solution Solve 3x + 6x – 9 = 6 x – 1 x - 1

Example #7 – Extraneous Solution Solve 2n + n + 3 = 1 1 – n n2 - 1 Example #8 – Solve 3 = x + 2 x - 1 x – 1