Graphs
● Factorising quadratic expression gives information about the graph of the
function
● X-intercepts found from numbers in each bracket
● Y-intercept is product of numbers in brackets
● If there is a single negative x in the factorised expression, the graph is
inverted
Example
Sketch the curve y = x2 + 5x + 6
x2 + 5x + 6 = (x + 2)(x + 3)
x – intercepts at x =- 2 and x =- 3
y – intercept at y = 6
Disguised Quadratics
● Some graphs don’t look like quadratics
● Can be rewritten as conventional-looking quadratics
Example
Solve x4 – x2 – 6 = 0
Let y = x2
y2 – y – 6 = 0
(y – 3)(y + 2) = 0
y = 3, y = – 2 → x2 = 3, x2 = – 2
x = ± √3
Turning Points
● Turning point called the vertex
● Find the turning point by rewriting quadratic function in different form
● Rewrite quadratic in completed square form → a(x – p)2 + q
● The coordinates of the turning point are (p, q)
Example
F ind the vertex of the function y = x2 + 6x + 11
x2 + 6x + 11 = (x + 3)2 + 11 – 9
= (x + 3)2 + 2
V ertex is at (- 3, 2)
The Quadratic Formula
If the quadratic function is too difficult to factorise then plug into the
quadratic formula
● Discriminant ( b2 – 4ac ) tells us about the nature of the x-intercepts:
○ If the discriminant is positive, there are two real solutions
○ If the discriminant is zero, there is only one real solution
○ If the discriminant is negative, there are no real solutions
● Useful to quickly calculate the discriminant first so you know whether there
are real solutions (ie whether there is any point in trying to solve the
function)