Quadratic Functions

1. Quadratic Functions The parent function is given as The graph of a quadratic function is called a parabola. This is the parent graph of all quadratic functions.

2. Quadratic Functions (-3,9) (3,9) A table of values can be constructed from the graph as given to the right. x y -3 9 -2 4 -1 1 (-2,4) (2,4) 0 0 1 1 2 4 3 9 (1,1) (-1,1) (0,0)

3. Quadratic Functions All other quadratic functions can be expressed in the form: (-3,9) (3,9) This is called the standard form. The general form is given as: (-2,4) (2,4) (1,1) (-1,1) (0,0)

4. Quadratic Functions (h,k) In standard form, h k (h,k) identifies the vertex of the parabola.

5. Quadratic Functions (1,8) In standard form, (5,8) a a affects the direction the parabola opens and how wide or narrow it will open. Since a=2 and it is positive, the parabola opens up and (2,2) (4,2) the y-values are all 2 times larger than on the parent graph. (3,0) 2

6. Quadratic Functions (3,0) In standard form, a (4,-2) (2,-2) If a is negative, the parabola will open down. Since a=-2 and it is negative, the parabola opens down and - 2 - 2 - 2 (5,-8) (1,-8) - 2 the y-values are all 2 times larger than on the parent graph. - 2 - 2 - 2 - 2 - 2

7. Quadratic Functions (3,8) The points where the parabola intersects the x-axis are called the Rootsor Zeros of the function. (4,6) (2,6) These roots occur when the y-value is equal to zero. Solving for x we get the values: (5,0) 5 1 (1,0) 1 5 X= X= 1 5 1 1 5 5 1 1 1 5 5 5

8. Quadratic Functions Example: Graph (5, -2) The vertex is (5, -2) (5, -2) (5, -2) (5, -2) The graph opens upward because 3 is positive. (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) The y-values are multiplied by 3. (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) up 3 (5, -2) (5, -2) Over 1

9. The zeros of the function can be found by setting y=0. Quadratic Functions Now solve for x. The roots or zeros are: (4.33, 0) and (5.66,0)