Sometimes it is enough to plot a set of data pairs to identify which function could be a good model. If this is not enough, we can be more precise by analyzing the data since each type of function has distinctive characteristics. We will review these characteristics for each function and explain how to determine which model to use.

When Is a Linear Function the Best Model?

In a linear function, each time the

x -

variable increases by

1

the function increases by a constant factor.

y = m x + b y = m (x + 1) + b ⇕ y = m x + b + m

Therefore, for equally spaced

x -

values of a set of data pairs, if the

y -

values have a common difference, then a linear function fits the model. Similarly, if the difference between pairs of

y -

values is approximately constant, then a linear function is a good model as well.

When Is a Exponential Function the Best Model?

For an exponential function, each time the

x -

variable increases by

1

the function is multiplied by a constant factor.

y = a b^{x} y = a b^{x + 1} ⇕ y = a b^{x} (b)

Therefore, for equally spaced

x -

values of a set of data pairs, if the

y -

values have a common ratio, then an exponential function fits the model. Similarly, if the ratio between pairs of

y -

values is approximately constant, then an exponential function is a good model as well.

When Is a Quadratic Function the Best Model?

Quadratic functions' data pairs have a characteristic feature as well. For equally spaced $x -$ values, the $y -$ values have constant second differences.

Similarly, if in the same conditions the second differences are approximately constant, then a quadratic function is a good model as well.

Exercise

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