UNIT 2: FACTOR QUADRATIC EXPRESSIONS UNIT 2 By the end of this unit, I will be able to: o Represent situations using quadratic expressions in one variable o Expand and simplify quadratic expressions in one variable o Factor quadratic expressions in one variable using a variety of methods o Solve quadratic equations by selecting and applying an appropriate factoring method o Determine and describe the connection between the factors of a quadratic equation and the x-intercepts of the graph o Explain any restrictions on the domain and range of a quadratic function in contexts arising from real-world applications o Express the equation of a quadratic function in standard form Name: 1
2.1 EQUATION FORMS Success Criteria: I can - Analyze and evaluate different forms of the quadratic function Warm up: Find f(2) for the following functions: f(x) = 2x 2 + 12x + 10 f(x) = 2(x + 3) 2 8 f(x) = 2(x + 1)(x + 5) What do you notice? A quadratic function can be expressed in three different forms: Standard Form - f(x) = ax 2 + bx + c Vertex Form - f(x) = a(x h) 2 + k Factored Form - f(x) = a(x r)(x s) What information can we get from each form by inspection: Properties Direction of Opening Standard Form f(x) = ax 2 + bx + c Vertex Form f(x) = a(x h) 2 + k Factored Form f(x) = a(x r)(x s) Maximum or Minimum Vertex Axis of Symmetry Max/Min Value y-intercept x-intercept(s) 2
For the quadratic functions below, identify the following key features. EX 1 Standard Form: f(x) = 2x 2 + x 3 Direction of Opening: Maximum or Minimum: y-intercept: EX 2 Vertex Form: f x = 2 x + 1 2 6 Direction of Opening: Maximum or Minimum: Vertex: Axis of Symmetry: Max/Min Value: y-intercept: EX 3 If we are given a function in factored form, we are able to find more key features - Factored form: f x = (x 1)(x 5) Direction of Opening: Maximum or Minimum: x-intercepts: Axis of Symmetry: Vertex: Max/Min Value: y-intercept: 3
Practice Find the key features for the functions below: EX 1 Standard Form: f x = 5x 2 + 8x + 74 Direction of Opening: Maximum or Minimum: y-intercept: EX 2 Vertex Form: f x = 3 x 4 2 + 2 Direction of Opening: Maximum or Minimum: Vertex: Axis of Symmetry: Max/Min Value: y-intercept: EX 3 Factored form: f x = (x + 6)(x 4) Direction of Opening: Maximum or Minimum: x-intercepts: Axis of Symmetry: Vertex: Max/Min Value: y-intercept: Homework: 4
2.2 GRAPHING FROM FACTORED FORM Success Criteria: I can Sketch the graph of a quadratic function in factored form Expand and simplify quadratic expressions in one variable Express the equation of a quadratic function in standard form Graphing from Factored Form A quadratic function can be graphed from factored form, y = a(x r)(x s) by: 1. Plotting the x-intercept(s) (x =, x = ) 2. Averaging the x-intercept(s) to find the axis of symmetry which occurs midway between the x-intercept(s) (x = h) 3. Substituting the x-value from (2) into the equation to find the max/min value (y = k) 4. Finding the y-intercept by letting x=0 and plotting it 5. Plotting the vertex (h, k) and connecting the points EX 1: Graph each quadratic function using the x-intercepts, vertex, and y-intercept. a) y = ½(x 3)(x + 5) b) f(x) = -x(x + 6) 5
Writing Equations in Standard Form A quadratic function in vertex or factored form can be converted into standard form by expanding and simplifying. This can be useful, as we know the c in standard form is the. f(x) = a(x h) 2 + k f(x) = a(x r)(x s) f(x) = ax 2 + bx + c EX 2: Write each quadratic function in standard form. Then, state the key features. a) f(x) = 4x(x 2) b) y = 2(x + 3) 2 9 c) f(x) = -(x + 2)(x 4) y-intercept: y-intercept: y-intercept: x-intercepts: x-intercepts axis of symmetry: axis of symmetry: axis of symmetry: vertex: vertex: vertex: Minimum or maximum? Minimum or maximum? Minimum or maximum? * Now we know how to convert from factored & vertex to standard, but how do we convert from standard to factored? 6
2.3 FACTORING POLYNOMIALS Success Criteria: I can Determine the greatest common factor for general expressions 7
2.4 FACTORING QUADRATIC FUNCTIONS Day 1 Success Criteria: I can Factor quadratic expressions in one variable using a variety of methods To factor a polynomial means to write it as a product. The opposite of factoring is expanding. factoring expanding 4(x + y) 4x + 4y There are several methods of factoring: A) COMMON FACTORING *Always try 1 st - Find the GCF for all terms (biggest # and variable that divides into all the terms) - Remove the GCF (by dividing each te X 1: Factor by common factoring: a) 4x+ 16x 2 b) 12m 2 + 15mn+ 3m c) 10x 3 25x 2 d) 6x 2 y 18xy + 9y 8
coefficient of 1 B) SIMPLE TRINOMIAL FACTORING, x 2 + bx + c - Find two numbers that add to b and multiply to c - Write as (x + ) (x + ) EX 2: Factor a) x 2 + 5x + 6 b) x 2 x 20 c) 2x 2 22x + 36 C) DIFFERENCE OF SQUARES, a 2 b 2 - Identify perfect squares o A perfect square is a number found by squaring another number. o E.g. 1, 4, 9, 25, 36, 49, 64, 81, 100, - Write as (a b)(a + b) EX 3: Factor a) x 2 4 b) 9m 2 16 9
c) y 2 + 100 d) 3p 2 75 Practice: Factor the following: 1) x 2 + 8x + 7 2) 3n 2 30n + 27 3) g 2 + 144 4) 9y 2-16 Homework: 10
2.5 FACTORING QUADRATIC FUNCTIONS Day 2 Success Criteria: I can Factor quadratic expressions in one variable using a variety of methods D) FACTORING TRICKY TRINOMIALS, ax 2 + bx + c (a 1) Replace middle term with two terms that add to b and multiply to ac; Group and common factor each pair of terms Remove common binomial factor EX 4: Factor a) 3x 2 + 14x + 8 b) 2m 2 + m 15 c) 4y 2 7y + 3 d) 6n 2 16y + 8 11
Recall: A perfect square is a number found by squaring another number. E.g. 1, 4, 9, 25, 36, 49, 64, 81, 100, E) PERFECT SQUARE TRINOMIALS, a 2 ± 2ab + b 2 - Identify perfect squares - Write as (a ± b) 2 Square of a binomial EX 6: Factor Twice product of square roots of first and last terms a) y 2 + 10y + 25 b) x 2 8xy + 16y 2 c) 4k 2 + 12k + 9 d) y 2 8y - 16 e) 16t 2 40t + 25 f) 49x 2 + 28xy + 4y 2 Homework: 12
Factoring Practice When choosing what factoring strategy to use, always start with GCF and work from there. Remember some expressions will be non factorable, meaning you cannot factor them. 1. 35m 3 + 5m 2. n 3 10n + 9 3. 5a 3 125 4. 3b 3 2b 5 5. 36x 3 81 6. m 3 + 2m 17 7. 16b 3 40b + 25 8.10x 3 + 100x + 250 9. 4y 3 15y 25 10. 36x B + 6x 13
Factoring: More Practice a) 7x 2 700 b) 5x 2 + 10x 40 c) 12x 2 20x d) 4x 2 9x - 9 e) 2x 2 18x + 36 f) -9x 2 15x + 6 g) 3x 2 6x + 12 h) 25x 2 9 i) 6x 2 + 15x 14
2.6 SOLVING QUADRATIC EQUATIONS BY FACTORING Success Criteria: I can Solve quadratic equations by selecting and applying an appropriate factoring method Determine and describe the connection between the factors of a quadratic equation and the x-intercepts of the graph Explain any restrictions on the domain and range of a quadratic function in contexts arising from real-world applications To solve a quadratic function means to find the,, or. To find these, we: For example, y = x 2 4 Find the x-intercepts: To prove that these are the zero(s) or root(s) of this equation, sub the x-intercepts into the original equation. If y=0, these are the zero(s) or root(s) of this equation. EX 1: Solve by factoring to find the x-intercept(s). a) y = 2x 2 14x b) f(x) =-3x 2 12x + 36 c) y = 2x 2 5x 12 d) f(x) = x 2 10x + 25 15
Solving quadratic equations is useful in application questions. EX 2: A stone is tossed from a bridge. Its height above the water is modeled by the equation, h(t) = - 5t 2 + 5t + 60, where h(t) is the height of the stone above the water, in metres, and t is the time, in seconds. Find the zero(s) of the function and explain their significance. EX 3: A relief package is released from a helicopter at 1600 feet. The height of the package can be modeled by the equation h(t) = -16t 2 + 1600, where h is the height of the package in feet and t is the time in seconds. The pilot wants to know how long it will take for the package to hit the ground. 16
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Practice: Sama is kicking a soccer ball on the top Emily Carr to his cousin who is on the soccer field. The height of the ball as a function of time is given by h(t)= -5t 2 +5t+10, where t is the time, in seconds, and h(t) is the height of the stone above the ground, in metres, at time t. a) Find the zeros of the function and explain their significance. b) Use the x-intercepts to find the vertex. What does it represent in this scenario? c) What is the height of the school? How do you know? d) Sketch a graph of the parabola: Homework: 18