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

Characteristics of Quadratics Line (axis) of symmetry - the line about which the parabola is symmetrical. The graphs of all parabolas have the same general U-shape. Vertex – the lowest (minimum) or highest (maximum) point of a parabola.

Quadratic Functions (-3,9) (3,9) A table of values can be constructed to graph a quadratic function. 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)

Quadratic Functions All quadratic functions can be expressed either in vertex form: (-3,9) (3,9) (-2,4) or in standard form: (2,4) (-1,1) (1,1) (0,0)

Quadratic Functions (h,k) When quadratic equations are written in vertex form, h k (h,k) identifies the vertex of the parabola. The line of symmetry is x = h

Quadratic Functions (1,8) In vertex form, (5,8) a a affects the direction the parabola opens and how wide or narrow it will open. In this problem, a=2 and it is positive, so the parabola opens up and (2,2) (4,2) (3,0) the y-values are all vertically stretched by a factor of 2. 2

Quadratic Functions (3,0) In vertex 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 vertically stretched by a factor of 2. - 2 - 2 - 2 - 2 - 2

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

The zeros of the function can be found by setting y=0. Quadratic Functions Now solve for x. The roots or zeros are: (1, 0) and (5,0)

Quadratic Functions Example: Graph The vertex is (5, -2) (5, -2) The graph opens upward since 3 is positive. (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) Pick x-values to the left and right of the vertex and plug into the equation to plot points. (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2) (5, -2)

The zeros of the function can be found by setting y=0. Quadratic Functions Now solve for x. The roots or zeros are: (4.184, 0) and (5.816,0)