warm-up x = 6 x = 4 x = 6 (x + 5)(x + 5) x+5 xx2x2 5x +55x25 x 2 + 10x + 25 (a + b)(a + b) = a 2 +...
TRANSCRIPT
Warm-Up
6353 x
19357 xx
242 xx x = 6
x = 4
x = 6
(x + 5)(x + 5) x +5
x x2 5x
+5 5x 25x2 + 10x + 25
(a + b)(a + b) = a2 + 2ab + b2
Homework Check
Page 144, # 18 page 144, # 24
Page 148, # 24 page 148, # 30
Warm-up
Simplify the fraction
114
98. 2
45
60.
3 16 0. x2
Solve.
4 12 0. x x2
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
X f(x) = 1/x Y
x can not be zero
Any restrictions on the domain?
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
f x = 1x
X f(x) = 1/x Y
5
0.5
0.2
0
-0.2
-0.5
-5
X f(x) = 1/x Y
5 1/5 0.2
0.5 1/0.5 2
0.2 1/0.2 5
0 1/0 ??
-0.2 -1/0.2 -5
-0.5 -1/0.5 -2
-5 -1/5 -0.2
Graphs of Rational Expressions Have students graph: f(x) = 5/(2x)
X f(x) = 1/2x YX f(x) = 1/(2x) Y
5
0.5
0.2
0
-0.2
-0.5
-5 Any restrictions on the domain?
x can not be zero
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
g x = 1
2x
f x = 1x
X f(x) = 1/(2x) Y
5 1/10 0.1
0.5 1/1 1
0.2 1/0.4 2.5
0 1/0 ??
-0.2 -1/0.4 -2.5
-0.5 -1/1 -1
-5 -1/10 -0.1
Graphs of Rational Expressions
Look at Geometer’s SketchPad for additional examples and explanation
Graphs of Rational ExpressionsSummary: 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 ExpressionsHomework:
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
18
60
2
3
x
x
3
10x
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 Expressionx x
x
2
2
6
3
Factor the top
x x
x
( )
6
3 2Cross out the common factor x.
( )x
x
6
3
3. Simplify a Rational Expression3
4 2
x
x x Factor the bottom
3
4
x
x x( ) Cross out the
common factor x.
3
4( ) x
4. Simplify a Rational Expression
5 10
15
2x x
x
Factor the top
5 1 2
15
x x
x
( )Cross out the
common factors of 5 and x.
( )1 2
3
x
5. Simplify a Rational Expression
x
x
2 16
3 12
Factor the top and bottom
( )( )
( )
x x
x
4 4
3 4Cross out the common factor (x + 4)
( )x 4
3
Divide Rational Expressions
(2x3 + 8x2 – 6x) ÷ 2x
x
xxx
2
682 23
x
xxx
2
)34(2 2
342 xx
Recognize Opposite Factors
When you have opposite factors, you will have to factor out a negative so that you can cancel.
6. Opposite Factors1
2 32
x
x x
Factor the bottom
( )
( )( )
1
3 1
x
x x
(1 – x) on the top and (x – 1) on the bottom are opposites. Factor out a negative so they will cancel.
( )
( )( )
x
x x
1
3 1 1
3( )x
7. Opposite Factorsx x
x
2 6 8
4
Factor the top
( )( )
( )
x x
x
4 2
4
(x – 4) on the top and (4 – x) on the bottom are opposites. Factor out a negative so they will cancel.
( )( )
( )
x x
x
4 2
4
( )( )
xx
2
11 2
x 2
Simplifying Rational ExpressionsSummary 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 = b2
Subtract a2 from each side
ab – a2 = b2 – a2
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
Giving a = 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 ExpressionsExample - Do what?
Factor
“Multiply”
Cancel
Answer – x can not
be -2, -3, or -4
127
3
2
232
2
xx
x
x
xx
)3)(4(
3
2
)1)(2(
xx
x
x
xx
)3)(4)(2(
)3)(1)(2(
xxx
xxx
)4(
)1(
x
x
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),
)1)(2(
)2)(2)(1)(1(
xx
xxxx
Divide Rational ExpressionsExample - Do what?
Invert Divisor
Factor
Multiply
Cancel
Answer – x can not
be -1 or 2 or -2
4
1
2
12
2
x
x
x
x
)1(
)2)(2(
2
)1)(1(
x
xx
x
xx
)2)(1( xx
1
4
2
1 22
x
x
x
x
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: http://www.doe.k12.ga.us/• 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)
x2 – 5x and x2 – 3x – 10 x(x – 5) and (x – 5)(x + 2) Answer: x(x – 5)(x + 2)
Add / Subtract Rational Expressions
xx
x 2
8
14
Solve and state the restrictions
x can not be 0
8
82
8
14
xx
x
x
x
8
1614
xx
x
8
16
8
14
x
x
8
174
Add / Subtract Rational Expressions
1
2
12
xx
x
Solve and state the restrictions
Restrictions:
1
2
)1)(1(
xxx
x
1
1
)1(
2
)1)(1(
x
x
xxx
x
)1)(1(
)1(2
xx
xx
)1)(1(
22
xx
xx
)1)(1(
23
xx
x
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
Divide (4y by (y - 3)2 10 6y )
4 10 6
3
2y y
y
( )
First one goes on top, second one goes on bottom.
2 2 1 3
3
( )( )
( )
y y
y
GCF and Long factor the top
2 2 1( )y 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 (x3 – 2x2 – 5x + 6), x = -2
Roots are x = -2, 1, and 3 (x3 – 3x2 – 6x + 8), x = 4
Roots are x = -2, 1, and 4 (x3 – 11x2 + 7x + 147), x = -3
– Roots are x = -3 and 7 duplicity two (2x3 – 10x2 - 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 x2, 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:
157
xx
Coordinate Geometry
Find any points of intersection for the graphs of the function and the function x
y8
2xy
Homework
Page 178, # 4 – 9 all and 13 – 15 all