Quadratics
Unit 5b
Quadratics
and
Rational Expressions
Name: ___________________________ Period: ___________
College Bound Math Teacher: _________________
Reference Sheet
|Axis of symmetry: |[pic] |
|Quadratic Formula: |[pic] |
|Determinate of a 2×2 matrix |[pic] |
|Inverse of a 2×2 matrix |[pic]( [pic] |
|I = interest B = Balance P = principle |
|r = rate (as a decimal) n = number of compounding periods t = time in years |
|Exponential growth & decay |B = P(1 + r)t B = P(1 – r)t |
|Simple Interest |Compound Interest |Continuous Interest |Future Value – periodic |Present Value – one time |Present Value - periodic |
|[pic] |[pic] |[pic] |[pic] |[pic] |[pic] |
|Key Words: |Key Words: |Key Words: |Key Words: |Key Words: |Key Words: |
|Simple interest |Compounded |Compounded |Total balance |Deposit now |Deposit now |
| |Annually, semiannually, |continuously |Each/every |Starting principal |Starting principal |
| |quarterly, etc. | | |goal |Each/every |
| | | | | |goal |
|Trigonometric Ratios |Sin A = [pic] Cos A = [pic] Tan A = [pic] |
|Coordinate Geometry |[pic] |
|Slope – Intercept Formula |y = mx + b |
|Law of Sines |[pic] |
|Law of Cosines |(side) [pic] |
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| |(angle) [pic] |
|Area of a Triangle |[pic] |
[pic]
Lesson# 1: Characteristics of Quadratics
The student will be able to understand the properties of a quadratic function.
The student will be able to determine values of a quadratic based on its properties
A.) Equation
The equation of a quadratic function is always in the form: ____________________________.
B.) Shape
The shape of a quadratic is called a ________________________.
|If the a-value is positive, the parabola looks like this: |If the a-value is negative, the parabola looks like this: |
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C.) Vertex - Minimum or Maximum?
If the parabola opens upward it has a __________________________________.
If the parabola opens downward it has a _________________________________.
This point (x, y) where it turns is called the ____________________________.
D.) Symmetry
A parabola has a line of symmetry called the __________________________________. Parabolas are symmetrical, or reflections of themselves down a vertical line (x = #).
Examples:
[pic]
E.) Roots/Zeros
The roots/zeros of a quadratic function are where that parabola crosses the ________________.
Parabolas can have:
|2 Zeros/Roots |1 Zero/Roots (tangent to) |0 Zeros/Roots |
|[pic] |[pic] |[pic] |
|Example 1: |Example 2: |
|[pic] |[pic] |
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|A = _______ B = _______ C = ________ |A = _______ B = _______ C = ________ |
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|Axis of symmetry: x = ________ |Axis of symmetry: x = ________ |
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|Vertex: ( ______ , ______ ) |Vertex: ( ______ , ______ ) |
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|Min or max? _____________________ |Min or max? _____________________ |
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|Zero(s)? _________________________________ |Zero(s)? _________________________________ |
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|Example 3: |Example 4: |
|[pic] |[pic] |
| |A = _______ B = _______ C = ________ |
|A = _______ B = _______ C = ________ | |
| |Axis of symmetry: x = ________ |
|Axis of symmetry: x = ________ | |
| |Vertex: ( ______ , ______ ) |
|Vertex: ( ______ , ______ ) | |
| |Min or max? _____________________ |
|Min or max? _____________________ | |
| |Zero(s)? _________________________________ |
|Zero(s)? _________________________________ | |
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|Example 5: |Example 6: |
|[pic] |[pic] |
|A = _______ B = _______ C = ________ |A = _______ B = _______ C = ________ |
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|Axis of symmetry: x = ________ |Axis of symmetry: x = ________ |
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|Vertex: ( ______ , ______ ) |Vertex: ( ______ , ______ ) |
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|Min or max? _____________________ |Min or max? _____________________ |
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|Zero(s)? _________________________________ |Zero(s)? _________________________________ |
|Example 7: |Example 8: |
|[pic] |[pic] |
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|A = _______ B = _______ C = ________ |A = _______ B = _______ C = ________ |
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|Axis of symmetry: x = ________ |Axis of symmetry: x = ________ |
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|Vertex: ( ______ , ______ ) |Vertex: ( ______ , ______ ) |
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|Min or max? _____________________ |Min or max? _____________________ |
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|Zero(s)? _________________________________ |Zero(s)? _________________________________ |
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|Example 9: |Example 10: |
|[pic] |[pic] |
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|A = _______ B = _______ C = ________ |A = _______ B = _______ C = ________ |
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|Axis of symmetry: x = ________ |Axis of symmetry: x = ________ |
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|Vertex: ( ______ , ______ ) |Vertex: ( ______ , ______ ) |
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|Min or max? _____________________ |Min or max? _____________________ |
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|Zero(s)? _________________________________ |Zero(s)? _________________________________ |
Homework #1 Characteristics of a quadratic function
|1: |2: |
|[pic] |[pic] |
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|A = _______ B = _______ C = ________ |A = _______ B = _______ C = ________ |
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|Axis of symmetry: x = ________ |Axis of symmetry: x = ________ |
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|Vertex: ( ______ , ______ ) |Vertex: ( ______ , ______ ) |
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|Min or max? _____________________ |Min or max? _____________________ |
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|Zero(s)? _________________________________ |Zero(s)? _________________________________ |
|3: |4: |
|[pic] |[pic] |
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|A = _______ B = _______ C = ________ | |
| |A = _______ B = _______ C = ________ |
|Axis of symmetry: x = ________ | |
| |Axis of symmetry: x = ________ |
|Vertex: ( ______ , ______ ) | |
| |Vertex: ( ______ , ______ ) |
|Min or max? _____________________ | |
| |Min or max? _____________________ |
|Zero(s)? _________________________________ | |
| |Zero(s)? _________________________________ |
Lesson #2a: Solving Quadratic Equations by Factoring
You have learned how to solve quadratic equations by graphing and using the quadratic formula. Another method used to solve quadratic equations is to factor (remember polynomials?!?!) and use the Zero Product Property.
Zero Product Property
|Words |Numbers |Algebra |
|If the product of two quantities equals zero, then one |3(0) = 0 |If ab = 0, |
|(or both) of the quantities must be equal to zero. |0(4) = 0 |Then a = 0 or b = 0 |
Use the Zero Product Property to solve each equation.
|Example #1: (x – 3)(x + 7) = 0 |Example #2: (x)(x – 5) = 0 |
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Solve each quadratic equation by factoring.
|Example #3: x2 + 7x + 10 = 0 |Example #4: x2 + 2x – 8 = 0 |
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|Example #5: x2 – 6x + 9 = 0 |Example #6: 3x2 – 4x + 1 = 0 |
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Classwork/practice:
[pic]
[pic]
Solve by factoring:
[pic]
[pic]
Homework #2a : Solve each of the following by factoring. SHOW ALL WORK!!!
1.) GCF
2.) Simple Trinomial
3.) Grouping
[pic]
[pic]
[pic]
Regents Practice: (Quadratic factoring)
|1 What is the solution set of the equation [pic]? |2 What is the solution set of the equation [pic]? |
|1) |1) |
|[pic] |[pic] |
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|2) |2) |
|{2,8} |[pic] |
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|3) |3) |
|[pic] |[pic] |
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|4) |4) |
|[pic] |[pic] |
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|3 The solution to the equation [pic] is |4 The solution set for the equation [pic] is |
|1) |1) |
|0, only |[pic] |
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|2) |2) |
|6, only |[pic] |
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|3) |3) |
|0 and 6 |[pic] |
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|4) |4) |
|[pic] |[pic] |
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|5 What is the solution set of [pic]? |6 What is the solution set of the equation [pic]? |
|1) |1) |
|[pic] |[pic] |
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|2) |2) |
|[pic] |[pic] |
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|3) |3) |
|[pic] |[pic] |
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|4) |4) |
|[pic] |[pic] |
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|7 What is the solution set for the equation [pic]? |8 What is the solution set of the equation [pic]? |
|1) |1) |
|[pic] |[pic] |
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|2) |2) |
|[pic] |[pic] |
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|3) |3) |
|[pic] |[pic] |
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|4) |4) |
|[pic] |[pic] |
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|9 The solution set of the equation [pic] is |10 The solution set for the equation [pic] is |
|1) |1) |
|[pic] |[pic] |
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|2) |2) |
|[pic] |[pic] |
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|3) |3) |
|[pic] |[pic] |
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|4) |4) |
|[pic] |[pic] |
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|11 The solutions of [pic] are |12 What is the positive solution of the equation [pic]? |
|1) | |
|[pic] and [pic] | |
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|2) | |
|2 and 14 | |
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|3) | |
|[pic] and [pic] | |
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|4) | |
|4 and 7 | |
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|13 Solve for x: [pic] |14 Solve for x: [pic] |
Lesson #2b: Formation of the Quadratic Equation, given the roots.
[pic]
[pic] [pic]
Homework #2b: Formation of the Quadratic Equation from the given roots
[pic]
[pic] [pic]
Lesson #3a: The Quadratic Formula
Students will be able to substitute values into the quadratic formula
Students will be able to use the quadratic formula to get rational results.
Sometimes the zeros on a graphed quadratic function are not easily identified, and so the Quadratic Formula can come in handy. (If you don’t like graphing, the quadratic formula will work EVERY time so it’s just another option!)
|Example #1: x2 -2x – 3 = 0 |Example #2: 2x2 + 3x – 5 = 0 |
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|Example #3: 9x2 – 6x + 1 = 0 |Example #4: 3x2 = -10x – 2 |
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Homework #3a: Quadratic Formula
Using the graph:
[pic]
Using the Quadratic Formula:
[pic]
Lesson #3b: Solving Quadratic Equations by the Quadratic Formula
Objectives:
• To be able to solve quadratic equations using the quadratic formula.
• To be able to solve applied problems using the quadratic formula.
Why Use the Quadratic Formula?
• Try to solve by factoring: [pic]
Consider the following situation: A person springs off a diving platform that is 9.8 meters from the surface of the water at a velocity of 22.3 meters per second. This can be modeled by the equation [pic], where s(t) is the height above the surface in meters and t is in seconds. According to the equation, when will the diver reach the surface of the water? At what height is the diver after 2 seconds?
Quadratic Formula
|Example 1: [pic] |Example 2: [pic] |
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|Example 3: [pic] |Example 4: [pic] |
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Homework #3b: Using the Quadratic Formula
[pic]
[pic]
Applied Problems
11. Revisit the problem presented at the beginning of the lesson:
Lesson #4: Quadratic Word Problems
Students will be able to solve word problems that require factoring a quadratic.
1) Find three consecutive odd integers such that the square of the first is equal to the second plus twice the third.
2) The measure of the length of a rectangle is 5 meters more than the width. If the area of the rectangle is 36 square meters, find the dimensions of the rectangle.
3) A garden is in the shape of a square. The length of one side of the garden is increased by 3 feet and the length of an adjacent side is increased by 2 feet. The garden now has an area of 72 square feet. What is the measure of a side of the original square garden?
4) The square of a positive number decreased by 4 times the number is 12. Find the positive number.
5) The length of a rectangle is 7 more than the side of a square. The width is equal to the side of the square. The area of the square is 56 less than the area of the rectangle. Find the width of the rectangle.
6) Mr. Walden has two square flower gardens. A side of the larger garden is 3 feet more than a side of the smaller garden. The sum of the areas of the two gardens is 269 feet. Find the length of a side, in feet, of each garden.
7) Find four consecutive positive integers such that the product of the first and fourth is four less than twice the first multiplied by the fourth.
8) Find three positive consecutive even integers such that the product of the first and second is 8 more than 38 times the third.
9) The length of a rectangle is four more than one-half the width of the rectangle. If the perimeter is fifty, find the area of the rectangle.
10) The sum of the squares of two consecutive positive integers is 85. Find the integers.
Part 2 –
Rational Expressions
[pic]
Lesson #5a: Undefined Fractions
The students will be able determine when a rational expression is undefined.
Where have you heard the word “undefined” before? In what math contexts?
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A rational expression is an algebraic equation whose numerator and denominator are polynomials. The value of the polynomial expression in the denominator can NEVER BE ________________ since division by zero is __________________.
This means that rational expressions may have excluded values (values that cannot be used because the denominator would be equal to zero).
**** You may need to factor!!!!!****
Find the excluded value(s) (values that make the denominator zero) of each rational expression.
|1.) [pic] |2.) [pic] |3.) [pic] |
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|4.) [pic] |5.) [pic] |6.) [pic] |
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Class work/ Homework #5a:
1 Which value of x makes the expression [pic] undefined?
|1) |[pic] |
|2) |[pic] |
|3) |3 |
|4) |0 |
2 The expression [pic] is undefined when the value of x is
|1) |[pic], only |
|2) |[pic] and 3 |
|3) |3, only |
|4) |[pic] and 2 |
3 Which value of n makes the expression [pic] undefined?
|1) |1 |
|2) |0 |
|3) |[pic] |
|4) |[pic] |
4 For which value of x is [pic] undefined?
|1) |[pic] |
|2) |0 |
|3) |3 |
|4) |4 |
5 The expression [pic] is undefined when x is
|1) |[pic], only |
|2) |2, only |
|3) |[pic] or 2 |
|4) |[pic], [pic], or 2 |
6 The function [pic] is undefined when the value of x is
|1) |0 or 3 |
|2) |3 or -3 |
|3) |3, only |
|4) |-3, only |
7 The algebraic expression [pic] is undefined when x is
|1) |0 |
|2) |2 |
|3) |3 |
|4) |9 |
8 Which value of x makes the expression [pic] undefined?
|1) |-5 |
|2) |2 |
|3) |3 |
|4) |-3 |
9 For which set of values of x is the algebraic expression [pic] undefined?
|1) |[pic] |
|2) |[pic] |
|3) |[pic] |
|4) |[pic] |
10 For which values of x is the fraction [pic] undefined?
|1) |1 and [pic] |
|2) |2 and [pic] |
|3) |3 and [pic] |
|4) |6 and [pic] |
11 A value of x that makes the expression [pic] undefined is
|1) |[pic] |
|2) |[pic] |
|3) |3 |
|4) |5 |
Lesson #5b: Simplifying Rational Expressions
The students will be able to simplify rational expressions
A rational expression is in simplest form when the numerator and denominator have no common factors besides 1. Remember that to simplify fractions you can divide out common factors that appear in both the numerator and the denominator. You can do the same to simplify rational expressions.
YOU CAN ONLY CROSS OUT GROUPS (in parentheses) OR SINGLE TERMS THAT REDUCE!!!
Simplify the following rational expressions by factoring. State the excluded values.
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|1.) [pic] |2.) [pic] |
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|3.) [pic] |4.) [pic] |
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|5.) [pic] |6.) [pic] |
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|7.) [pic] |8.) [pic] |
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Homework #5b: Simplifying Practice – Simplify each rational expression. State any excluded values.
[pic]
[pic]
Lesson 5 a & b Extra Practice if needed
[pic]
[pic]
Lesson #6: Multiplying Rational Expressions
The students will be able to multiply and simplify rational expressions.
Recall: How do we multiply fractions? _____________________________ But you can simplify FIRST!!!
[pic]
We can do the same with rational expressions – cross cancel common factors.
|1.) [pic] |2.) [pic] |
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|3.) [pic] |4.) [pic] |
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|5.) [pic] |6.) [pic] |
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Classwork: Multiplying Rational Expressions – Multiply and simplify – state any excluded values.
[pic]
[pic]
Homework #6: Multiplying Rational Expressions
[pic]
Lesson #7: Dividing Rational Expressions
Recall: How do we divide fractions?
[pic]
We do the same when dividing rational expressions – multiply by the reciprocal. (COPY CHANGE FLIP)
|1.) [pic] |2.) [pic] |
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|3.) [pic] |4.) [pic] |
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|5.) [pic] |6.) [pic] |
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Multiplying/ Dividing Rational Expressions –
Multiplying: STRAIGHT ACROSS – simplify and cross out on top/bottom
Dividing: multiply by the reciprocal, then simplify.
1 What is the product of [pic] and [pic] expressed in simplest form?
|1) |x |
|2) |[pic] |
|3) |[pic] |
|4) |[pic] |
2 What is the product of [pic] and [pic] expressed in simplest form?
|1) |[pic] |
|2) |[pic] |
|3) |[pic] |
|4) |[pic] |
3 Express the product of [pic] and [pic] in simplest form.
4 Perform the indicated operation and express in simplest form: [pic]
5 Perform the indicated operation and express in simplest form: [pic]
6 Express the product in simplest form: [pic]
7 Perform the indicated operations and express in simplest form: [pic]
8 Express the product in simplest form: [pic]
9 If the length of a rectangular garden is represented by [pic] and its width
is represented by [pic], which expression represents the area of the garden?
|1) |[pic] |
|2) |[pic] |
|3) |[pic] |
|4) |[pic] |
More practice of Multiplying & Dividing Rational Expressions.
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
[pic]
Even More Multiplication & Division Practice
[pic]
[pic]
[pic]
[pic]
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[pic]
[pic]
[pic]
[pic]
These are factorable so you will get rational results
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