A mathematical model is an equation or graph that represents a real-world situation and can be used to analyze and make predictions about that situation. Mathematical models may be exact or approximate.

Because real-world scenarios usually involve many variables, we often identify the most important ones and making reasonable assumptions about the rest. A good model simplifies the situation as much as possible without significantly reducing the accuracy of its predictions.

In a mathematical model, constants and coefficients are called parameters. The general shape of a model is given by its family (linear, quadratic, exponential, etc.), but the more specific values (like intercepts, asymptotes, or steepness) are controlled by the parameters.

A linear model is represented by a straight-line graph.

Since a linear model can be defined by one point and a gradient or two points, they are the simplest models to construct. The most common form of a linear model is y=ax+b, where a is the slope and b is the y-intercept.

Extrapolation is when we predict values beyond the domain of the given points. Extrapolating may work for certain situations, but it does not work for many others. Pay attention to the context of a model when extrapolating and consider whether the observed behavior is likely to change in the long-run.

Your understanding of extrapolation can be tested by questions that ask you to interpret plausible inputs and outputs.

We use a piecewise linear model when different linear models apply over different parts of the domain of points. Basically, a piecewise linear model is a collection of smaller, domain-restricted models.

We write piecewise functions with the following notation:

A quadratic model has a turning point (vertex) at which its minimum or maximum value occurs. The general form of a quadratic equation is ax2+bx+c.

If a<0, the turning point of a quadratic is its maximum; if a>0, the turning point of a quadratic is its minimum.

The roots of a quadratic correspond to the x-intercepts of its graph. When x=a or x=β, the entire expression equals zero, which is reflected on the graph.

The equation of the parabola below is −(x−α)(x−β):

The graph of a quadratic function has the general shape of a parabola.

It is symmetrical about the axis of symmetry and has a maxima or minima at the vertex, which lies on the axis of symmetry.

An exponential model represents quantities that multiply repetetively by a constant factor b. The basic form of an exponential is bx, but any exponential can be written in the form Abx+k.

The graph of an exponential model is a curve that approaches a horizontal asymptote at y=k on one side, and has a y-intercept at (0,A+k). Because of the asymptote on an exponential graph, exponential models are good at describing behaviors that level off over time.

Exponential growth describes quantities that increase by the same factor over a certain amount of time. Algebraically, exponential growth are functions of the form

where b>1. b is called the growth factor.

Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is positive.

Stewart EJ, Madden R, Paul G, Taddei F (2005), CC BY-SA 4.0

Exponential decay describes quantities that decrease by the same factor over a certain amount of time. Exponential decay are functions of the form

where 0<b<1. b is called the decay factor.

Note: Aekt is another model for exponential growth if the instantaneous growth rate, k, is negative.

Cubic models have the form ax3+bx2+cx+d. Cubic graphs may have 0 or 2 turning points. When cubic graphs have 0 turning points, they have a short flat section where the function appears constant.

Sinusoidal models describe quantities that repeat in regular intervals, or periodically, and are typically of the form y=asin(bx)+c or y=acos(bx)+c.

A sinusoidal curve y=acos(bx)+c is graphed below with key features.

The principal axis, the line around which the sinusoid oscilates, is given by y=c.

The amplitude, or the maximum distance the sinusoid reaches above and below the principal axis, is a.

The period, or the horizontal distance between consecutive maxima, is given by b360​° (or b2π​rad for HL).