Some Common Functions and their Graphs – Quadratic Functions

quadratic functions of x A function in the form y = ax2 + bx + c or f(x) = ax2 + bx + c, where a, b and c are constants and a  0, is called a quadratic function of x. Example: (i) y = –2x2 + x quadratic functions of x (ii) f(x) = (x + 3)2 + 5

Graph of y = ax2 + bx + c (a > 0) axis of symmetry The graph has reflectional symmetry about . y x opens upwards for a > 0 a vertical line c The graph and its axis of symmetry intersect at . y-intercept = c a point x-intercepts vertex (It is the minimum point of the graph.) minimum value of y

Graph of y = ax2 + bx + c (a < 0) axis of symmetry y x vertex (It is the maximum point of the graph.) maximum value of y opens downwards for a < 0 x-intercepts c y-intercept = c

Coordinates of the vertex = (–1, 0) Can you find the axis of symmetry and the coordinates of the vertex of the graph of y = x2 + 2x + 1? axis of symmetry x y 4 3 2 1 1 2 8 6 4 2 y = x2 + 2x + 1 Axis of symmetry: x = –1 vertex Coordinates of the vertex = (–1, 0)

Follow-up question Consider the graph of y = (x + 1)(5 – 2x). (a) Determine the direction of opening. (b) Find the x-intercept(s) and the y-intercept of the graph. y = (x + 1)(5 – 2x) (a) y = 5x + 5 – 2x2 – 2x ∴ y = –2x2 + 3x + 5 ∵ Coefficient of x2 = –2  0 ∴ The graph opens downwards.

+ (b) ∵ y = –2x2 + 3x + 5 ∴ The y-intercept of the graph is 5. The x-intercepts of the graph are the roots of (x + 1)(5 – 2x) = 0. ) 2 5 )( 1 ( = - + x 2 5 or 1 = - + x 2 5 or 1 = - x ∴ The x-intercepts of the graph are –1 and . 2 5