Graph of quadratic functions

1. Graph of quadratic functions We start with a simple graph of y = x2. Important features  It is  shaped.  It is symmetrical about a line x = 0 (i.e. y axis).  It has a vertex at (0,0) (i.e. the minimum point). y y = x2 x Vertex(0, 0)

2. Graph of quadratic functions By changing the equation slightly, we can shift the curve around without changing the basic shape. The graph of y = x2 + 5 can be obtained by translating the graph of y = x2 five units in the y-direction. y y = x2 + 5 x Vertex (0, 5)

3. y x Graph of quadratic functions The graph of y = x2 – 10 can be obtained by translating the graph of y = x2ten unitsin the negative y direction. y = x2 - 10 Vertex (0, -10)

4. Graph of quadratic functions I we replace x by x – k in the equation of a graph then the graph produces a translation of k units in the x direction. The graph of y = (x – 2)2 can be obtained by translating the graph of y = x2 two units in the x direction. y y = (x – 2)2 x Vertex (2, 0)

5. Graph of quadratic functions In a similar fashion, the graph of y = (x + 4)2 is a shift of – 4 in the x-direction, the vertex is at (-4, 0). y y = (x + 4)2 x Vertex (-4, 0)

6. Graph of quadratic functions We start with a simple graph of y = -x2 Important features  It is  shaped.  It is symmetrical about a line x = 0 (i.e. y axis).  It has a vertex at (0,0) (i.e. the maximum point). y Vertex(0, 0) x y = - x2

7. y x Graph of quadratic functions We can also have combinations of these transformations: The graph of y = (x – 2)2 – 10 has a shift of 2 units in the x-direction and –10 in the y-direction, with minimum point at (2, -10). y = (x – 2)2 - 10 Vertex (2, -10)

8. Use of the discriminant b2 – 4ac The discriminat of the quadratic function y = ax2 + bx + c is the value of b2 – 4ac. Discriminat b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0 Number of roots: twoone None Intersection with the x-axis Two pointsTouch at one pointDo not meet Sketch a >0 Sketch a < 0