Identify first and second differences

For a relation given in a table or sequence with equally spaced input values, first differences compare consecutive output values, while second differences compare consecutive first differences; constant first differences indicate a linear relation, and constant nonzero second differences identify a quadratic relation. The learner connects the pattern to $y=ax^2+bx+c$ , recognizing that for unit-spaced inputs the second difference is $2a$ , and interprets its sign and size as evidence of the parabola’s direction and curvature. This treatment is limited to numerical tables with equally spaced inputs, not higher-order finite differences or irregular spacing.

Detailed Explanation: Identify first and second differences

When the input values are equally spaced, compare the output values in two stages.

First differences: Subtract each output from the next output.
Second differences: Subtract each first difference from the next first difference.

Consider this table:

$x$	$y$
$0$	$3$
$1$	$6$
$2$	$11$
$3$	$18$
$4$	$27$

The $x$ -values increase by $1$ each time, so the inputs are equally spaced.

Step 1: Find the first differences

Subtract consecutive $y$ -values:

6-3=3,\qquad 11-6=5,\qquad 18-11=7,\qquad 27-18=9

The first differences are:

3,\ 5,\ 7,\ 9

They are not constant, so the relation is not linear.

Step 2: Find the second differences

Now subtract consecutive first differences:

5-3=2,\qquad 7-5=2,\qquad 9-7=2

The second differences are:

2,\ 2,\ 2

Because the second differences are constant and nonzero, the relation is quadratic.

For unit-spaced inputs, the constant second difference equals $2a$ in

y=ax^2+bx+c.

Here,

2a=2 \quad\Rightarrow\quad a=1.

Since $a$ is positive, the parabola opens upward. The positive second difference also shows that the outputs are increasing at an increasing rate.

In fact, the table comes from

y=x^2+2x+3,

but finding the constant second difference is enough to identify the relation as quadratic and determine that $a=1$ .

Learn by doing: Identify first and second differences

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First and Second Differences - Values Table to Function Type

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