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The graph of a quadratic function is called a parabola. The parent function is given as y x 2 This is the parent graph of all quadratic functions. (-3,9) A table of values can be constructed from the graph as given to the (-2,4) right. (3,9) (2,4) (1,1) (-1,1) (0,0) y x 2 x -3 -2 -1 0 1 2 3 y 9 4 1 0 1 4 9 y x (-3,9) 2 All other quadratic functions can be expressed in the form: (3,9) y a (x h )2 k This is called the standard form. (-2,4) (2,4) The general form is given as: y ax 2 bx c (1,1) (-1,1) (0,0) (h,k) In standard form, y = a x h 2 + k (h,k) identifies the vertex of the parabola. (1,8) (5,8) In standard form, y = a x h 2 + k a affects the direction the parabola opens and how wide or narrow it will open. (2,2) (4,2) (3,0) y = 2 x 3 2 + 0 Since a=2 and it is positive, the parabola opens up and the y-values are all 2 times larger than on the parent graph. (3,0) (2,-2) In standard form, (4,-2) y = a x h 2 + k If a is negative, the parabola will open down. Since a=-2 and it is negative, the-parabola opens down 2 -2 - 2and (5,-8) (1,-8) - 2 the y-values are all 2 times 2 -2 larger than on the parent 2 y = - 2 -x2 3 2 + 0 graph. (3,8) (4,6) (2,6) The points where the parabola intersects the xaxis are called the Roots or Zeros of the function. These roots occur when the y-value is equal to zero. 0 2(x 3)2 8 (1,0) 1 1 1 Solving for x we get the (5,0) 5 values: 5 1 51 X= 1 15 5 X=5 5 2 y 2(x 3) 8 Example: Graph y 3(x 5) 2 2 (5, -2) The vertex is (5, -2) (5, -2) (5, -2) (5, -2) The graph opens (5, upward -2) (5, -2) (5, -2) because 3 is positive. (5, (5,-2) -2) (5, -2) (5, (5,-2) -2)-2) The y-values are (5, (5,(5, -2) multiplied by 3. -2) (5, -2) (5, -2)-2) (5, (5,(5, -2) -2) (5, (5,-2) -2) Over 1 up 3 The zeros of the function can be found by setting y=0. 0 3(x 5) 2 2 Now solve for x. 0 3(x 5)2 2 0 2 3(x 5)2 2 2 2 3(x 5)2 1 1 2 3(x 5)2 3 3 2 (x 5)2 3 2 (x 5)2 3 2 x 5 3 The roots or zeros are: 5 2 x 55 3 (4.33, 0) and (5.66,0) 5 2 x 55 3 2 x 3 5.66 x 5 4.33 x