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**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

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**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

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**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

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**Graph of a Quadratic Function**

f (x)=(x -1) - 4 (1,-4) vertex

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**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 ) then 3(x + 2x +1 ) (1) 3(x + 1)

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**Graph of a Quadratic Function**

y = 3x + 6x +2 Vertex(-1, -1) y = (x + 1) -1

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**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 =

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**The Quadratic function**

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

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**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

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**Identify the Vertex and Axis of Symmetry**

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

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**Steps for Graphing Quadratic Function**

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

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**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)

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**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

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**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

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**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

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**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

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