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

2. 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.

3. Now let us see the graphs of quadratic functions

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

5. 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.

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

7. 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)

8. 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).

9. 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

10. 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

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

12. 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

13. 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.

14. 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

15. 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

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

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

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

19. 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?

20. 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