Graphs of Quadratic Function • Introducing the concept: Transformation of the Graph of y = x2 4 fx = x2 -5 2 5 Graph of f(x) = 2 ax and 2 a(x-h) • 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) = ax2 + bc + c 0. Graphs of QUADRATIC FUNCTIONS are called PARABOLAS. Now let us see the graphs of quadratic functions Graph of QUADRATIC FUNCTION 4 fx = -x2 gx = -2x2 hx = -0.5x2 fx = x2 2 gx = 2x2 2 LINE , OR AXIS OF SYMMETRY -5 hx = 0.5x2 -5 5 VERTEX 5 -2 VERTEX -2 LINE , OR AXIS OF SYMMETRY -4 • 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) = ax2, we know the following about its a graph. Compared with the graph of f(x) = x2. 2. If a > 1, the graph is stretched vertically. a < 1, the graph is shrunk vertically. 3. If a < 0, the graph is reflected across the x-axis. 1. If EXAMPLE: a. Graph f(x) =3x2 b. Line of Symmetry? Vertex? fx = 3x2 4 2 -5 5 LINE OF SYMMETRY The y-axis VERTEX -2 (0,0) Exercise: 2 a. Graph f(x) = -1/4 x b. Line of symmetry and Vertex? • Your answer should be like this fx = - x2 4 LINE OF SYMMETRY 2 Y-AXIS VERTEX (0,0) -5 5 -2 -4 In f(x) = ax2, 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). Compare the Graph of f(x) = 2(x+3)2 to the graph of f(x) = 2x2. 4 4 fx = 2x+32 LINE OF SYMMETRY, X = -3 VERTEX (0,0), fx = 2x2 2 2 SYMMETRY, Y-AXIS VERTEX (0,3) -5 5 -5 5 EXAMPLE: 2 a. Graph f(x) = - 2(x-1) b. Line of Symmetry and Vertex? fx = -2x-12 2 VERTEX (h, 0) = (1,0) -5 LINE OF SYMMETRY, X=1 5 -2 -4 -6 EXERCISES: a. Graph f(x) = 3(x-2)2 b. Line of Symmetry and Vertex? 6 fx = 3x-2 2 4 2 -5 LINE OF SYMMETRY, X=2 VERTEX (2,0) 5 Graph of f(x) = 2 a(x-h) +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 Guidelines for Graphing Quadratic Functions, f(x)=a(x-h)2 + k • 1. 2. 3. 4. When graphing quadratic function in the form f(x)=a(x-h)2+k, The line of symmetry is x-h=0, or x = h. The vertex is (h,k). If a > 0, then (h,k) is the lowest point of the graph, and k is the MINIMUM VALUE of the function. If a < 0, then (h,k) is the highest point of the graph, and k is the MAXIMUM VALUE of the function. 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? fx = 2x+3 2-2 4 LINE OF SYMMETRY, X=-3 2 -5 5 VERTEX: ( -3,-2) -2 MINIMUM: -2 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 Answer #1 6 fx = 3x-2 2+4 VERTEX: (2,4) 4 2 MIN: 4 LINE OF SYMMETRY:X =2 -5 5 Answer #2 fx = -3x+22-1 4 VERTEX: (-2,-1) 2 MAX: -1 LINE OF SYMMETRY:X = -2 -5 5 -2 -4 ANALYZING f(x) = a(x-h)2+k • Objective: Determine the characteristics of a function f(x) = a(x-h)2+k 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? Answer in #1 and #2 a. What is the Vertex? b. Line of Symmetry? #1. (1/4, -2) #2. ( -5, 7) X=¼ X = -5 c. Is there a Min / Minimum. The Maximum. The Max Value? graph extends graph extends upward since 3>0 downward since –4<0. d. What is the Min.Value is –2 Max.Value is 7 min / max value?