Quadratic functions are polynomials of degree 2 with the general form f(x)=ax2+bx+c. The discriminant, ∆ =b2-4ac, determines the number of x-intercepts. Cubic functions are polynomials of degree 3 with the general form f(x)= ax3+bx2+cx+d. Quartic functions are polynomials of degree 4 with the general form f(x)=ax4+bx3+cx2+dx+e. Higher degree polynomial functions have more complex behaviors determined by their factors and coefficients.

1. Maths Methods
Types of Functions 1

2. Types of Functions 1
Quadratic functions
Polynomial functions of degree 2
General quadratic equation is:
f(x)=ax2+bx+c
The discriminant, ∆ =b2-4ac, determines the
number of x-intercepts on a quadratic graph
<0, none; =0, one; >0, two

3. Types of Functions
You can use the quadratic formula to find x-
intercepts:
Quadratic formula:
The equation of the axis of symmetry is x= -
b/2a
The turning point is (-b/2a, c-(b2/4a))

4. Types of Functions
The factorised form of the quadratic function
is:
f(x)=a(x-b)(x-c)
x-intercepts are at: (b,0) and (c,0)
y-intercepts is: a x -b x -c
The turning point is halfway between the x-
intercepts
If a>0 minimum Turning Point
If a<0 maximum Turning Point

5. Types of Functions
Cubic functions:
Polynomials of degree 3
General cubic equation is:
f(x)= ax3+bx2+cx+d
If a>0, the graph is positive
If a<0, the graph is negative

6. Types of Functions
Factorised forms of cubic equations:
y=a(x-b)(x-c)(x-d)
a is the dilation from the x-axis
b,c, and d are the x-intercepts
If a<0, the graph is reflected about the x-axis

7. Types of Functions
Factorised forms of cubic equations:
y=(x-b)2(x-c)
a is the dilation from the x-axis
b and c are the x-intercepts
b is the x-intercept and the x-coordinate of the
turning point

8. Types of Functions
Power form of cubic equations:
y=(x-b)3
b is the x-intercept and the x-coordinate of the
point of inflection

9. Types of Functions
Quartic functions:
Polynomials of degree 4
General quartic equation:
f(x)=ax4+bx3+cx2+dx+e
a>0, the graph is positive
a<0, the graph is negative

10. Types of Functions
Quartic functions in factorised form:
f(x)=a(x-b)(x-c)(x-d)(x-e)
x-intercepts at x=b,c,d and e

11. Types of Functions
Quartic functions in factorised form:
f(x)= a(x – b)(x – c)(x – d)(x – e)
x-intercepts at b, c, d and e
f(x)=a(x – b)(x – c)3
Cubed factor (x – c)3 shows stationary point of
inflection at x=c
f(x)=a(x – b)2(x – c)2
Repeated factors (x – b)2 and (x – c)2 show
stationary points at x=b and x=c
f(x)=ax2(x – b)(x – c)
Repeated factor x2 indicates a stationary point at
x=0