Graph of quadratic functions We start with a simple graph of y = x 2. y = x 2 x y Vertex(0, 0) 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).

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

Graph of quadratic functions The graph of y = x 2 – 10 can be obtained by translating the graph of y = x 2 ten units in the negative y direction. x y y = x Vertex (0, -10)

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. x y The graph of y = (x – 2) 2 can be obtained by translating the graph of y = x 2 two units in the x direction. y = (x – 2) 2 Vertex (2, 0)

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). x y y = (x + 4) 2 Vertex (-4, 0)

Graph of quadratic functions We start with a simple graph of y = -x 2 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 = - x 2 Vertex(0, 0) 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). x y y = (x – 2) Vertex (2, -10)

Use of the discriminant b 2 – 4ac The discriminat of the quadratic function y = ax 2 + bx + c is the value of b 2 – 4ac. Discriminat b 2 – 4ac > 0b 2 – 4ac = 0b 2 – 4ac < 0 Number of roots:two one None Intersection with the x-axis Two points Touch at one point Do not meet Sketch a >0 Sketch a < 0