Graphs of Quadratic Function

1. Graphs of Quadratic Function • Introducing the concept: Transformation of the Graph of y = x2

2. Graph of f(x) = ax2 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) = ax2 + 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 VERTEX LINE , OR AXIS OF SYMMETRY

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) = ax2, we know the following about its graph. Compared with the graph of f(x) = x2. • If > 1, the graph is stretched vertically. • If < 1, the graph is shrunk vertically. • If a < 0, the graph is reflected across the x-axis.

6. EXAMPLE:a. Graph f(x) =3x2b. Line of Symmetry? Vertex? LINE OF SYMMETRY The y-axis VERTEX (0,0)

7. Exercise:a. Graph f(x) = -1/4 x2b. Line of symmetry and Vertex? • Your answer should be like this LINE OF SYMMETRY Y-AXIS VERTEX (0,0)

8. 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).

9. Compare the Graph of f(x) = 2(x+3)2 to the graph of f(x) = 2x2. LINE OF SYMMETRY, X = -3 VERTEX (0,0), SYMMETRY, Y-AXIS VERTEX (0,3)

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

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

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, • 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.

14. Example:a. Graph f(x) = 2(x+3)2 – 2b. 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+42. g(x) = -4x+5)2+7

20. Answer in #1 and #2