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MPM2D Name:__TEACHER___ Quadratics Review – Day 1 of 2 Quadratic Relations: Vocabulary Finite differences: are the differences between consecutive values of the dependent variable. • First difference: are the values you get when you do the finite differences only once. If they are all equal, the relation is linear. • Second difference: are the values you get when you do the finite differences twice. If they are all equal, the relation is quadratic. Quadratic: is the relationship you have when the second differences are constant. It is a second degree polynomial. Degree: is the highest exponent that appears in any term of the expanded form of a polynomial. Properties of Quadratic Relations A parabola: is the shape of the graph of a quadratic relation. A vertex: is the point on the graph of a parabola with the greatest y-coordinate (if the graph opens down) or the least y-coordinate (if the graph opens up). It is usually represented by the coordinates (h, k). Optimal value: is the y-coordinate of the vertex when a quadratic relation is used to model a situation. Maximum: is the greatest y value of the quantity being modeled, when the parabola opens down. Maximum is (2,-2) Minimum: is the smallest y value of the quantity being modeled, when the parabola opens upwards. Minimum is (3,2) Axis of symmetry: is the vertical line that passes through the vertex. It is the perpendicular bisector of the segments joining any two points on the parabola that have the same y-coordinates. Axis of symmetry for top is x=3 Axis of symmetry for bottom is x=2 Zeros or x-intercepts: are the points where the parabola crosses the x axis. 3.4 The Role of the Zeros of a Quadratic Relation The factored form of a quadratic relation is y = a(x – r)(x – s), provided that a ≠ 0. a>0 If a > 0, the parabola opens up and has a minimum. maximum • If a < 0, the parabola opens down and has a maximum. • minimum The zeros of a quadratic relation are the solutions to 0 = a ( x − r )( x − s ) a<0 The x-coordinate of the vertex can be found by using the midpoint formula with the values of the zeros. r+s x= 2 The y-coordinate of the vertex can be found by substituting the x coordinate into the equation of the relation. The value of a can be determined by substituting the coordinates of a point on the curve and the zeros of the relation. Ex. If y = (x + 2)(x – 4), determine the zeros and the coordinates of the vertex. *****Example to be done on the board Zeros ⇒ 0 = (x + 2)(x – 4) x + 2 = 0 or x – 4 = 0 x = –2 Vertex ⇒ −2+4 2 x =1 x= x=4 ∴ the zeros are –2 and 4 y = (x + 2)(x - 4) y = (1 + 3)(1 − 4 ) y = (4 )(− 3) y = −12 ∴ The coordinates of the vertex are (1,−12 ) Ex. What is the equation of a parabola whose zeros are 2 and –3, if it passes through the point A(4,28)? Zeros mean y = a(x – 2)(x + 3) Point A means 28 = a(4 – 2)(4 + 3) 28 = a(2)(7) 28 = 14a 28 =a 14 a=2 ∴ Equation is y = 2(x – 2)(x + 3) Standard Form of a Quadratic Relation Standard form or expanded form, y = ax2 + bx + c, can be obtained from the factored form by using the distributive property and then simplifying. Ex. Expand the expression y = (2x + 3)(3x – 7). = 2x(3x – 7) + 3(3x – 7) = 6x2 – 14x + 9x – 21 = 6x2 – 5x – 21 3.8 Factoring Quadratic Expressions Factoring means writing a quadratic relation that was in the standard form y = ax2 + bx + c and expressing it as a product of two binomial factors. Strategy: 1. Find two numbers, n and m, such that: a. Their product equals the product of a and c: n × m = a × c b. Their sum equals b: n + m = b 2. Replace the b in the standard form of the equation with n + m 3. Group the first two terms together and the second two terms together and take a common factor out of each term. 4. Remove the common factor from the two groups of terms. Ex. Factor x2 – 14x + 45 Factor 4a2 – 12a + 9 Difference of squares is a special quadratic made up of a square at the beginning minus a square at the end. It looks like a2x2 – b2 and is factored into (ax + b)(ax – b). Perfect square is a special quadratic made up of a square at the beginning, a square at the end and two times the product of their roots in the middle. It looks like a2x2 + 2abx + b2 and is factored into (ax + b)(ax + b) or (ax + b)2. Ex. Factor t2 – 8t + 16 Ex. Factor 25x2 – 9