Warm-Up x = 6 x = 4 x = 6 (x + 5)(x + 5) x+5 xx2x2 5x +55x25 x x + 25 (a + b)(a + b) = a 2 + 2ab + b 2

Homework Check Page 144, # 18page 144, # 24 Page 148, # 24page 148, # 30

Warm-up Simplify the fraction Solve.

Standards for Rational Functions MM1A2c. Add, subtract, multiply, and divide polynomials. MM1A2d. Add, subtract, multiply, and divide rational expressions. MM1A3d. Solve simple rational equations that result in linear equations or quadratic equations with leading coefficient of 1.

Rational Expressions Essential questions: 1. What is a rational expression? 2. What doe the graph look like, and how does it move? 3. How do we simplify rational expressions?

11.3 Simplifying rational expressions A fraction whose numerator and denominator are nonzero polynomials

Real Life Applications of Rational Expressional Calculate concentrations Optimize perimeter / area Maximize profit / time Minimize Cost / Piece Average cost (cost / # pieces) Monthly payment per month Calculate piece part rates

Graphing Rational Expressions Discuss domain, range, restrictions (has to be based on original expressions),

Graphs of Rational Expressions Have students graph the parent curve Xf(x) = 1/xY x can not be zero Any restrictions on the domain? Xf(x) = 1/xY Xf(x) = 1/xY 51/ / / /0?? / / /5-0.2

Graphs of Rational Expressions Have students graph: f(x) = 5/(2x) Xf(x) = 1/2xY Xf(x) = 1/(2x)Y Any restrictions on the domain? x can not be zero Xf(x) = 1/(2x)Y 51/ / / /0?? / /1 -5-1/10-0.1

Graphs of Rational Expressions Look at Geometer’s SketchPad for additional examples and explanation

Graphs of Rational Expressions Summary: Can not have values of x that make the denominator equal zero (the domain is restricted) Multiplying the parent function by a number x > 1 stretches graph vertically. Multiplying the parent function by a number 0 < x < 1 shrinks graph vertically. Adding a constant shifts the graph up or down Multiplying by -1 reflects across the x- axis

Graphs of Rational Expressions Homework: Pg 153, # 1 – 11 all

Simplifying Rational Expressions Factor each numerator and denominator Cross out the terms common to the numerator and denominator Keep track of domain restrictions based on the original equation

When a rational expression’s numerator and denominator have no factors in common (other than 1)

1. Simplify a Rational Expression Reduce the numbers and subtract the exponents. Where the larger one is, is where the leftovers go to keep exponents positive

2. Simplify a Rational Expression Factor the top Cross out the common factor x.

3. Simplify a Rational Expression Factor the bottom Cross out the common factor x.

4. Simplify a Rational Expression Factor the top Cross out the common factors of 5 and x.

5. Simplify a Rational Expression Factor the top and bottom Cross out the common factor (x + 4)

Divide Rational Expressions (2x 3 + 8x 2 – 6x) ÷ 2x

Recognize Opposite Factors When you have opposite factors, you will have to factor out a negative so that you can cancel.

6. Opposite Factors Factor the bottom (1 – x) on the top and (x – 1) on the bottom are opposites. Factor out a negative so they will cancel.

7. Opposite Factors Factor the top (x – 4) on the top and (4 – x) on the bottom are opposites. Factor out a negative so they will cancel.

Simplifying Rational Expressions Summary Review Factor each numerator and denominator Cross out the terms common to the numerator and denominator Keep track of domain restrictions based on the original equation

Let’s practice!! Pg 163, # 3 – 15 by 3’s and 16 – 19 all (9 problems)

Let:a = b Multiply each side by b:ab = b 2 Subtract a 2 from each side ab – a 2 = b 2 – a 2 Factor each side:a(b – a) = (b – a)(b + a) There is a (b – a) on each side, so cancel: a = b + a But a = b was given, so by substitution: b + a = a + a = 2a Givinga = 2a Divide each side by a:1 = 2 Warm-Up – What is wrong?

Have someone do problem 18 on page 163 (from last night’s homework) on the board. What is the value of x to make the perimeter equal the area?

Multiply Rational Expressions Factor both numerators and denominator “Multiply” numerators “Multiply” denominators Cancel like factors in numerator and denominator Discuss domain restrictions (has to be based on original expressions),

Multiply Rational Expressions Example - Do what? Factor “Multiply” Cancel Answer – x can not be -2, -3, or -4

Divide Rational Expressions Invert the divisor (turn up side down) Factor both numerators and denominators “Multiply” numerators “Multiply” denominators Cancel like factors in numerator and denominator Discuss domain, range, restrictions (has to be based on original expressions),

Divide Rational Expressions Example - Do what? Invert Divisor Factor Multiply Cancel Answer – x can not be -1 or 2 or -2

In Class Practice: Page 167, # 13 Page 167, # 16

Homework: Page 164, # 16 Page 167, # 3 – 18 by 3’s and 19 This is 8 problems

EOCT Review Place to start: Click: Testing, End of Course Tests (also has GHSGT reviews) Math 1 Released Items gives sample questions Math 1 Released items commentary gives answers and description to sample questions Student study guides gives example questions EOCT Released Tests are for Algebra, Geometry, not Math 1, but you can see what kind of questions they have asked in the past

Add / Subtract Rational Expressions We must have the same denominator to be able to add or subtract rational expressions. Can always find a common denominator by multiplying the denominators Make common denominator by multiplying each rational expression by “1”

Add / Subtract Rational Expressions The Least Common Denominator (LCD) of two or more rational expressions is the product of the factors in common (used only once) times the non-common factors.

Assuming these are denominators: Find the least common denominator (2x – 4) and (x – 2)  2(x – 2) and (x – 2)  Answer: 2(x – 2) x 2 – 5x and x 2 – 3x – 10  x(x – 5) and (x – 5)(x + 2)  Answer: x(x – 5)(x + 2)

Add / Subtract Rational Expressions Solve and state the restrictions x can not be 0

Add / Subtract Rational Expressions Solve and state the restrictions Restrictions: x can not be1 or -1

Practice Page 173, # 3 – 21 by 3’s and 22 and 23

Warm-Up Divide by long division: 5889 ÷ 23

Divide a Polynomial by a Binomial It is easy if you can factor the numerator and denominator

Divide Polynomials (if you can not factor Why do it? –To find factors, zeros, solutions, roots of the equation The “zeros” are the values of x that make the equation equal zero. Review long division Apply to polynomials If the value of x makes the expression equal zero, you found a root. If the remainder = zero, you found a root.

Divide a polynomial by a binomial First one goes on top, second one goes on bottom. GCF and Long factor the top Cancel common factor (y – 3)

Divide Polynomials Fold paper hot dog Show calculations on one side Show division on the other side

Divide Polynomials Verify the one root, then find the others (x 3 – 2x 2 – 5x + 6), x = -2  Roots are x = -2, 1, and 3 (x 3 – 3x 2 – 6x + 8), x = 4  Roots are x = -2, 1, and 4 (x 3 – 11x 2 + 7x + 147), x = -3 –Roots are x = -3 and 7 duplicity two (2x 3 – 10x 2 - 7x + 35), x = 5

Practice Pg 159, # 3 – 10 all (8 problems)

Solving Rational Equations Eliminate the denominator by multiplying by the common denominator. Solve: If x 2, x, and constant: –Move everything to one side of the equality sign, making everything equal zero –Factor –Use Zero Product Rule to solve. –Check your answers

Painting a Car: At a car body shop, Kayla needs 5 hours working alone to paint a car. It takes Emily 7 hours to paint the same car. Write rational expressions that represent the fractional parts of the work that Kayla and Emily complete when painting the car together.

Painting a Car: Write rational expressions that represent the fractional parts of the work that Kayla and Emily complete when painting the car together. Kayla - amount painted per hour: x/5 Emily – amount painted per hour: x/7

Painting a Car: Write and solve a rational equation to find the time it would take them to paint the car if they work together. The time to paint one car is:

Coordinate Geometry Find any points of intersection for the graphs of the function and the function

Homework Page 178, # 4 – 9 all and 13 – 15 all