Graphing non-linear functions (Higher Tier)

Understanding Non-Linear Functions

A non-linear function is a function whose graph doesn’t form a straight line.
Non-linear functions include quadratic functions, cubic functions, reciprocal functions, exponential functions and more.
The degree of a polynomial is the highest power of the variable that appears in the function. For instance, in a quadratic function (degree 2), the highest power of the variable is 2.

Quadratic Functions

A quadratic function is a type of non-linear function with the general formula: y = ax^2 + bx + c.
The graph of a quadratic function, known as a parabola, can open upwards (if a > 0) or downwards (if a < 0).
The vertex of the parabola is a key feature and represents the minimum or maximum point of the function depending on where it opens.
The roots or zeros of the function are the x-values where the graph crosses or touches the x-axis.

Cubic Functions

A cubic function is another type of non-linear function with the general formula: y = ax^3 + bx^2 + cx + d.
The graph of a cubic function can have a characteristic ‘S’ shape, or just a single bend.
Cubic functions can have one, two or three roots.

Reciprocal Functions

A reciprocal function has the general form y = a/x + c,
The graph of a reciprocal function consists of two hyperbolas located in the opposite quadrants.
Reciprocal functions have two asymptotes: the x-axis (y = 0) and the y-axis (x = 0).

Exponential Functions

An exponential function has the format y = ab^x, where ‘a’ is a constant, ‘b’ is the base and ‘x’ is the exponent.
The base, ‘b’, is usually a positive number, and when b > 1, the function increases and when 0 < b < 1, the function decreases.
The graph of an exponential function passes through the point (0,1).

Graphing Non-Linear Functions

Non-linear functions can be plotted on graph paper or using graphic display calculators or software.
When graphing, identify important features to display such as intercepts, asymptotes, maxima, minima, and turning points.
A smooth curve should be drawn through the plotted points, not a series of straight-line segments.

Applications of Non-Linear Functions

Non-linear functions, including quadratics, cubics and exponential functions, often depict real-world phenomena such as physics problems, biological growth models or financial calculations.
Understanding their properties, and how to represent them graphically, helps in analysing and predicting behaviours of these real-world applications.