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Algebra Notes: Quadratic Functions Quadratic Functions There are two different forms of quadratic functions that we will study. Standard/Expanded Form Factored Form Graphs of Quadratic Functions The graph of a quadratic function is called a … The vertex of a parabola is … The roots of a parabola are … Roots are also called … Unit Goals The goals of this unit are as follows: 1. To combine polynomials by addition, subtraction, and multiplication. 2. To translate quadratic expressions from expanded form to factored form and vice-versa. 3. To solve quadratic equations. 4. To graph quadratic functions. 5. To solve application problems involving quadratic functions. Quadratic Function Notes Page # 1 Algebra Notes: Quadratic Functions Polynomials Equations like are called polynomials. A Quadratic Polynomial is … Adding and Subtracting Polynomials When you combine two polynomials by addition or subtraction, you can only combine _________________. Like terms are … (2x3 + 3x2 – 4x + 7) + ( 7x4 – 8x3 + 3x2 + 9) Adding Polynomials (5x2 – 7x + 12) + (- 8x2 – 3x + - 8) Quadratic Function Notes Page # 2 Algebra Notes: Quadratic Functions Subtracting Polynomials (10x2 – 9x - 18) - (6x2 – 13x - 8) Another Example (3.63x2 – 2.15x – 9) – (- 5.21x2 + 6.01x – 3.14) Multiplying Binomials Three techniques that can be used to multiply binomials are… 1. 2. 3. FOIL (x + 2)(x + 4) Quadratic Function Notes Page # 3 Algebra Notes: Quadratic Functions Punnett Square (2x - 7)(-3x + 6) Vertical Multiplication (2x – 3)(3x + 4) Factoring Factoring is… (x + 4)(x + 1) = The Process of Factoring Write x2 + 14x + 24 in factored form A Little Tougher One Write x2 - 5x - 24 in factored form Quadratic Function Notes Page # 4 Algebra Notes: Quadratic Functions You Try These Write each of the following quadratic expressions in factored form. x2 + 8x + 15 = x2 - 12x + 35 = x2 + 8x - 20 = FACTORING ALGORITHM 1 Original Expression: + 1 ac = ____, ____ = b 2 3 ____ + ____ + ____ + ____ 4 ___ ( ___ + ___ ) + ___ ( ___ + ___ ) 5 ( ___ + ___ ) ( ___ + ___ ) FACTORING ALGORITHM 2 Original Expression: + ac = ____, ____ =b 1 2 3 ____ + ____ + ____ + ____ 4 ___ ( ___ + ___ ) + ___ ( ___ + ___ ) 5 ( ___ + ___ ) ( ___ + ___ ) FACTORING ALGORITHM 3 Original Expression: + 1 ac = ____, ____ = b 2 3 4 5 ____ + ____ + ____ + ____ ___ ( ___ + ___ ) + ___ ( ___ + ___ ) Quadratic Function Notes Page # 5 ( ___ + ___ ) ( ___ + ___ ) Algebra Notes: Quadratic Functions Flow Map Time! Let’s Try Some 3x2 - 13x + 12 = 12x2 + 24x – 15 = Quadratic Function Notes Page # 6 Algebra Notes: Quadratic Functions The Five Types of Problems That I Have to Solve in Quadratic Applications The equation y =-11.5x2 + 1009.825 x – 9095 describes the profit, y, that a company makes when it sells its product for a price of x dollars. Answer the following questions. 1. How should the company set its price in order to make the maximum profit? What is the maximum profit that the company can make? 2. How much profit will the company make if they set the price at $30? 3. What are the “break even” prices for the company? 4. How much money would the company lose if they gave their product away for free? 5. If the company needed their profit to be exactly $11,500 for tax purposes, what could they set their price at? Problem #1: Find the Vertex 1. How should the company set its price in order to make the maximum profit? What is the maximum profit that the company can make? Record the Keystrokes that you need to use to find the VERTEX on your calculator Problem #2: Find y, Given x 2. How much profit will the company make if they set the price at $30? Quadratic Function Notes Page # 7 Algebra Notes: Quadratic Functions Problem #3: Find the Roots 3. What are the “break even” prices for the company? Record the Keystrokes that you need to use to find the ROOTS on your calculator Problem #4: Find the Y-Intercept 4. How much money would the company lose if they gave their product away for free? Problem #5: Find x, Given y 5. If the company needed their profit to be exactly $11,500 for tax purposes, what could they set their price at? Quadratic Function Notes Page # 8 Algebra Notes: Quadratic Functions Three Methods for Solving Quadratic Equations 1. 2 3. Inverse Operations An Important Analogy We solve the equation 3x – 7 = 14 by using _______________ to _____________ the variable x. First, we “undo” the -7 by … Then, we “undo” the 3 by dividing … ___________________ undoes _______________________ ___________________ undoes _______________________ #1: Using Square Roots An Example Solve the following equation: 3 x 2 7 14 Quadratic Function Notes Page # 9 Algebra Notes: Quadratic Functions Try These 3 2 x 47 2 5 x 2 8 33 9 x 2 5 2 x 2 18 #2: Factored Form and Solving Equations If a·b=0, what must be true? This is called … Factored Form and Solving Equations (cont.) If (x+2)(x+4)=0, what must be true? Solve These Equations (x + 5)(x + 1) = 0 (x - 3)(x + 4) = 0 Quadratic Function Notes Page # 10 (2x - 3)(3x + 5) = 0 Algebra Notes: Quadratic Functions Solving By Factoring #1 Solve the equation x2 + 10x = - 21 by factoring. Step #1: Set the equation equal to 0. Step #2: Factor Step #3: Apply the Zero Product Property Solving By Factoring #2 Solve the equation 2x2 = 17x + 19 by factoring. Step #1: Set the equation equal to 0. Step #2: Factor 2 3 + 1 ac = ____, ____ = b Original Expression: ____ + ____ + ____ + ____ 4 ___ ( ___ + ___ ) + ___ ( ___ + ___ ) 5 ( ___ + ___ ) ( ___ + ___ ) Step #3: Apply the Zero Product Property Quadratic Function Notes Page # 11 Algebra Notes: Quadratic Functions Solve These Equations x2 + 7x + 12 = 0 2x2 + 5x = 12 5x2 - 4x - 4 = 2x2 - 15x #3: The Quadratic Formula An equation of the form _____________________________ can always be solved by using the quadratic formula. Example #1 Solve the equation 3x2 – 4x – 5 = 0 Step #1: Identify a, b, and c. Step #2: Plug a, b, and c into the formula Quadratic Function Notes Page # 12 Algebra Notes: Quadratic Functions Step #3: Simplify the expression Step #4: Evaluate the two answers Make sure to do a quick check using your calculator!!!!! Example #2 Solve the equation -4x2 + 3x = -6 Try These 2x2 + 5x - 12 = 0 5x2 - 4x - 4 = 2x2 - 15x Quadratic Function Notes Page # 13 Algebra Notes: Quadratic Functions Now that you have completed the investigating parabolas chart, take some time to analyze your results and answer the following questions. 1. How can you tell if a parabola “opens up” or “opens down” simply by looking at the equation? 2. How can you determine the location of the roots of a parabola before you actually graph the parabola? 3. In terms of the roots, where will the vertex of the parabola always be located? What other pattern do you notice related to the vertex? 4. How can you determine the location of the y-intercept of the parabola before you actually graph the parabola? Quadratic Function Notes Page # 14