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Graphs of Quadratic Function Introducing the concept: Transformation of the Graph of y = x 2

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Graph of f(x) = ax 2 and a(x-h) 2 Objective: Graph a function f(x)=a(x-h) 2, and determine its characteristics. Definition: A QUADRATIC FUNCTION is a function that can be described as f(x) = ax 2 + bc + c 0. Graphs of QUADRATIC FUNCTIONS are called PARABOLAS.

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Now let us see the graphs of quadratic functions

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Graph of QUADRATIC FUNCTION LINE, OR AXIS OF SYMMETRY VERTEX LINE, OR AXIS OF SYMMETRY VERTEX

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Thus the y-axis is the LINE SYMMETRY. The point (0,0) where the graph crosses the line of symmetry, is called VERTEX OF THE PARABOLA Next consider f(x) = ax 2, we know the following about its graph. Compared with the graph of f(x) = x 2. 1.If > 1, the graph is stretched vertically. 2.If < 1, the graph is shrunk vertically. 3.If a < 0, the graph is reflected across the x-axis.

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EXAMPLE: a. Graph f(x) =3x 2 b. Line of Symmetry? Vertex? LINE OF SYMMETRY The y-axis VERTEX (0,0)

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Exercise: a. Graph f(x) = -1/4 x 2 b. Line of symmetry and Vertex? Your answer should be like this LINE OF SYMMETRY Y-AXIS VERTEX (0,0)

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In f(x) = ax 2, let us replace x by x – h. if h is positive, the graph will be translated to the right. If h is negative the translation will be to the left. The line, or axis of symmetry and the vertex will also be translated the same way. Thus f(x) = a(x-h) 2, the axis of symmetry is x = h and the vertex is (h, 0).

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Compare the Graph of f(x) = 2(x+3) 2 to the graph of f(x) = 2x 2. VERTEX (0,3) LINE OF SYMMETRY, X = -3 VERTEX (0,0), SYMMETRY, Y-AXIS

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EXAMPLE: a. Graph f(x) = - 2(x-1) 2 b. Line of Symmetry and Vertex? VERTEX (h, 0) = (1,0) LINE OF SYMMETRY, X=1

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EXERCISES: a. Graph f(x) = 3(x-2) 2 b. Line of Symmetry and Vertex? LINE OF SYMMETRY, X=2 VERTEX (2,0)

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Graph of f(x) = a(x-h) 2 +k Objective: Graph a function f(x) = a(x-h) 2 + k, and determine its characteristics. In f(x) = a(x-h) 2, let us replace f(x) by f(x) – k f(x) – k = a(x-h) 2 Adding k on both sides gives f(x) = a(x-h) 2 + k. The Graph will be translated UPWARD if k is Positive and DOWNWARD if k is NEGATIVE. The Vertex will be translated the same way. The Line of Symmetry will NOT be AFFECTED

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Guidelines for Graphing Quadratic Functions, f(x)=a(x-h) 2 + k When graphing quadratic function in the form f(x)=a(x-h) 2 +k, 1.The line of symmetry is x-h=0, or x = h. 2.The vertex is (h,k). 3.If a > 0, then (h,k) is the lowest point of the graph, and k is the MINIMUM VALUE of the function. 4.If a < 0, then (h,k) is the highest point of the graph, and k is the MAXIMUM VALUE of the function.

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Example: a. Graph f(x) = 2(x+3) 2 – 2 b. Line of Symmetry, Vertex? c. is there a min/max value? If so, what is it? LINE OF SYMMETRY, X=-3 VERTEX: ( -3,-2) MINIMUM: -2

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Exercises: for each of the following, graph the function, find the vertex, find the line of symmetry, and find the min/ max value. 1. f(x) = 3(x-2) 2 + 4 2. f(x) = -3(x+2) 2 - 4

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Answer #1 VERTEX: (2,4) MIN: 4 LINE OF SYMMETRY:X =2

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Answer #2 VERTEX: (-2,-1) MAX: -1 LINE OF SYMMETRY:X = -2

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ANALYZING f(x) = a(x-h) 2 +k Objective: Determine the characteristics of a function f(x) = a(x-h) 2 +k

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EXAMPLE: Without graphing, find the vertex,line of symmetry, min/max value. Given: 1. f(x) = 3(x-1/4) 2 +4 2. g(x) = -4x+5) 2 +7 a. What is the Vertex? b. Line of Symmetry? c. Is there a Min / Max Value? d. What is the min / max value?

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Answer in #1 and #2 a. What is the Vertex? #1. (1/4, -2)#2. ( -5, 7) b. Line of Symmetry? X = ¼X = -5 c. Is there a Min / Max Value? Minimum. The graph extends upward since 3>0 Maximum. The graph extends downward since –4<0. d. What is the min / max value? Min.Value is –2Max.Value is 7

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