This topic focuses on recognising, sketching and interpreting several common non linear graphs. Each type has a characteristic shape that can be identified from its equation.

 

 

Quadratic Graphs of the Form y = ax² + b

Equations of this form produce quadratic graphs, which are curved and called parabolas.

 

The equation is written as
\( y = ax^2 + b \)

 

The value of a controls the direction of the curve.

a is positive so the graph opens upwards
a is negative so the graph opens downwards

 

The value of b gives the y intercept.

 

The turning point of the graph is always on the y axis at
\( (0, b) \)

 

Example
$$
y = 2x^2 - 3
$$

 

This graph opens upwards and crosses the y axis at \( -3 \).

 

 

Quadratic Graphs of the Form y = (ax + b)(cx + d)

This form also produces a quadratic graph, but it is written in factorised form.

 

The equation is written as
\( y = (ax + b)(cx + d) \)

 

The x intercepts can be found by setting each bracket equal to zero.

 

Example

$$
y = (x - 2)(x + 1)
$$

 

The graph crosses the x axis at

$$
x = 2
$$

 

and

$$
x = -1
$$

 

These intercepts help you sketch the curve accurately. The graph is still a smooth parabola.

 

 

Reciprocal Graphs of the Form y = a/x

Equations of this form produce reciprocal graphs.

 

The equation is written as
\( y = \frac{a}{x} \)

 

These graphs have two separate curves and never touch the axes.

 

The x axis and y axis act as asymptotes

 

a is positive so the graph lies in quadrants 1 and 3
a is negative so the graph lies in quadrants 2 and 4

 

Example

$$
y = \frac{3}{x}
$$

 

As \( x \) increases, \( y \) decreases but never reaches zero.

 

 

Cubic Graphs of the Form y = ax³

Equations of this form produce cubic graphs.

 

The equation is written as
\( y = ax^3 \)

 

These graphs pass through the origin and have a smooth S shape.

 

a is positive so the graph rises from bottom left to top right
a is negative so the graph falls from top left to bottom right

 

Example

$$
y = x^3
$$

 

Cubic graphs have rotational symmetry about the origin.

 

 

 

Key Points to Remember

Quadratic graphs are smooth curves called parabolas.
The sign of a controls whether quadratic and cubic graphs open upwards or downwards.
Factorised quadratics show x intercepts clearly from the brackets.
Reciprocal graphs never touch the axes and have asymptotes.
Cubic graphs pass through the origin and have an S shaped curve.

 

Recognising the form of an equation allows you to sketch and interpret non linear graphs accurately and efficiently.