1 Any function of the form y = f (x) = ax 2 + bx + c where a 0 is called a Quadratic Function 2 Example: y = 3x a = 3, Note that if function 2 - 2x + 1 b = -2, c = 1 a=0 we simply have the linear y = bx + c 3 Consider the simplest quadratic equation y=x2 Here a = 1, b = 0, c = 0 Plotting some ordered pairs (x, y) we have: y = f (x ) = x 2 x -3 -2 -1 0 1 2 3 f (x ) 9 4 1 0 1 4 9 (x, y ) (-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4) (3, 9) 4 (x, y) (-2, 4) 4 (-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4) y (2, 4) 3 2 y = x2 1 (3, 9) -3 -2 -1 Vertex (0, 0) 1 2 3 x A parabola with the y-axis as the axis of symmetry. 5 Graphs of y = ax 2 will have similar form and the value of the coefficient ‘a ’ determines the graph’s shape. y y = 2x 2 y=x 4 3 2 y = 1 /2 x 2 2 a>0 opening up -3 -2 1 -1 1 2 3 x 6 y x a<0 opening down y = -2x 2 In general the quadratic term ax 2 in the quadratic function f (x ) = ax 2 +bx + c determines the way the graph opens. 7 Consider f (x ) = ax 2 +bx + c In a general sense the linear term bx acts to shift the plot of f (x ) from side to side and the constant term c (=cx 0) acts to shift the plot up or down. a>0 Notice that c is the y -intercept where x = 0 and f (0) = c y x-intercept c a<0 c x y-intercept Note also that the x -intercepts (if they exist) 8 are obtained by solving: y = ax 2 +bx + c = 0 It turns out that the details of a quadratic function can be found by considering its coefficients a, b and c as follows: (1) Opening up (a > 0), down (a < 0) (2) y –intercept: c (3) x -intercepts from solution of y = ax 2 + bx + c = 0 You solve by factoring or the quadratic formula - b - b (4) vertex = ,f 2a 2a 9 Example: y = f (x ) = x 2 -x-2 here a = 1, b = -1 and c = -2 (1) opens upwards since a > 0 (2) y –intercept: -2 (3) x -intercepts from x 2 - x - 2 = 0 or (x -2)(x +1) = 0 x = 2 or x = -1 -b (4) vertex: h 2a 1 1 , f 2 2 1 1 , -2 4 2 10 y (-1, 0) -2 (2, 0) -1 0 -1 -2 1 2 y=x 2 x -x-2 1 1 , -2 4 2 -3 11 Example: y = j (x ) = x 2 -9 here a = 1, b = 0 and c = -9 (1) opens upwards since a > 0 (2) y –intercept: -9 (3) x -intercepts from x 2 - 9 = 0 or x 2 = 9 x = 3 (4) vertex at (0, -9) - b - b , j 2a 2a 12 y (-3, 0) -3 (3, 0) 0 y=x -9 x 3 2 -9 (0, -9) 13 Example: y = g (x ) = x 2 - 6x + 9 here a = 1, b = -6 and c = 9 (1) opens upwards since a > 0 (2) y –intercept: 9 (3) x -intercepts from x 2 - 6x + 9 = 0 or (x - 3)(x - 3) = 0 x = 3 only (4) vertex: -b h 2a 3, g 3 3, 0 14 y 9 (0, 9) y=x (3, 0) 3 2 - 6x + 9 x 15 Example: y = f (x ) = -3x 2 + 6x - 4 here a = -3, b = 6 and c = -4 (1) opens downwards since a < 0 (2) y –intercept: -4 (3) x -intercepts from -3x 2 + 6x - 4 = 0 (there are no x -intercepts here) (4) vertex at (1, -1) - b - b ,f 2a 2a Vertex is below x-axis, and parabola opens down! 16 y 1 (1, -1) -1 y = -3x -4 x 2 2 + 6x - 4 (0, -4) 17 The Quadratic Formula It is not always easy to find x -intercepts by factoring ax 2 + bx + c when solving ax 2 + bx + c = 0 Quadratic equations of this form can be solved for x using the formula: x - b b 4ac 2 2a 18 Example: Solve x 2 − 6x + 9 = 0 here a = 1, b = -6 and c = 9 - b b 4ac x 2a 2 Note: the expression inside the radical is called the “discriminant” - (-6) (-6) 4(1)(9) x 2(1) 2 6 0 Note: discriminant = 0 x one solution 2 6 2 3 only as found previously 19 Example: Solve x 2 -x-2=0 here a = 1, b = -1 and c = -2 x ( 1) ( 1) 4(1)( 2) 2(1) 2 1 9 13 x 2 2 1 3 13 x or x 2 2 x 1 or x 2 Note: discriminant > 0 two solutions 20 Example: Find x -intercepts of y = x 2 -9 Solve x 2 - 9 = 0 a = 1, b = 0, c = -9 x 0 x 4 ( 9 ) 2 36 6 2 2 Note: discriminant > 0 two solutions x = 3 or x = -3 21 Example: Find the x -intercepts of y = f (x) = -3x Solve -3x x x b 6 2 2 + 6x - 4 a = -3, b = 6 and c = -4 + 6x - 4 = 0 b2 4ac 2a 36 4( 3)( 4 ) 2( 3) 6 12 x ? 6 Note: discriminant < 0 no Real solutions 12 is undefined there are no x -intercepts as we discovered in 22 an earlier plot of y = -3x 2 + 6x - 4 The end. 23

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