1. Some Common Functions and their Graphs – Quadratic Functions

2. 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

3. 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

4. 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

5. 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)

6. 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.

7. + (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