The Graph of a Quadratic Function is called a Parabola The parabola opens up when a > 0 The parabola opens down when a < 0 When the parabola opens up the lowest point on the graph is called the Vertex When the parabola opens down the highest point on the graph is called the Vertex

The Graph of a Quadratic Function (cont) The Vertex of a quadratic function is a pair (x, y) where the x coordinate is sometimes denoted by h and the y by k. The function f (x) = ax +bx + c can be written in the form f (x) = a (x + h) + k by completing the square

Graphing the Quadratic Function Using Transformations Given the Quadratic function We complete the square and get f (x)=(x -1) – 4. Here h = 1 and k = -4 The graph is y = x shifted horizontal right 1 unit and vertically down 4 units. The result is given below

Graph of a Quadratic Function f (x)=(x -1) - 4 (1,-4) vertex

Graphing the Quadratic Function Using Transformations To find the vertex of the given function f (x) = 3x + 6x +2 We write f (x) as 3(x + 2x ) + 2 then 3(x + 2x +1 ) + 2 -3(1) 3(x + 1) - 1

Graph of a Quadratic Function y = 3x + 6x +2 Vertex(-1, -1) y = (x + 1) -1

The Graph of a Quadratic Function The vertical line which passes through the vertex is called the Axis of Symmetry or the Axis Recall that the equation of a vertical line is x =c For some constant c x coordinate of the vertex The y coordinate of the vertex is The Axis of Symmetry is the x =

The Quadratic function Opens up when a>o opens down a < 0 Vertex axis of symmetry

Identify the Vertex and Axis of Symmetry of a Quadratic Function Vertex =(x, y). thus Vertex = Axis of Symmetry: the line x = Vertex is minimum point if parabola opens up Vertex is maximum point if parabola opens down

Identify the Vertex and Axis of Symmetry Vertex x = y = Vertex = (-1, -3) Axis of Symmetry is x = = -1 = -3 = -1

Steps for Graphing Quadratic Function Method I Given the Quadratic f (x) = ax + bx + c = 0 a 0 1. Complete the square to write the function as f (x) = a (x – h) + k 2. Graph function in stages using transformation

Steps for Graphing Quadratic Function Method 2 Determine 1. the vertex 2. the Axis of Symmetry: the line x = 3. the y intercept. That is f(0)

Steps for Graphing Quadratic Function(cont.) 4. Determine the x–intercept, that is, f (x) = 0 a. If there are 2 x-intercepts and the graph crosses the x axis at 2 points b. If there is 1 x-intercept and the graph crosses the x axis at 1 point c. If there are no x-intercepts and the graph does not cross the x axis. 5. Use the Axis of Symmetry and y –intercept to get an additional point and plot the points

Finding the Maximum and Minimum Points If the parabola opens up, that is, if a > 0 the vertex is the lowest point on the graph and the y coordinate of the vertex is the minimum point of the quadratic function If the parabola opens down, that is, if a < 0 the vertex is the highest point on the graph and the y coordinate of the vertex is the maximum point of the quadratic function

Finding the Maximum and Minimum Points Page 153 # 60 Find the maximum or minimum f (x) = -2x + 12x a = -2 < 0. Thus the parabola opens down and has a maximum Vertex x = = = 3 y = f(3) = 18 Vertex = (3, 18) Maximum is y = 18

Finding the Maximum and Minimum Points f(x) = 4x – 8x + 3 a = 4 >0. Thus the parabola opens up and has a minimum Vertex (x = -(-8)/2(4) = 1, y=f(1)= -1) Vertex(1, -1) Minimum = y=f(1)= -1 Note the range is y -1