Recognize quadratic relations from tables

A quadratic relation is identified in a table of values by equal, nonzero second differences when the input values are equally spaced; its first differences change by a constant amount, unlike a linear relation, whose first differences are constant. The pattern indicates a relationship that can be represented by $y=ax^2+bx+c$ , with the second difference connected to the coefficient $a$ , and supports connecting tables with graphs and equations. More advanced polynomial tests and irregularly spaced inputs are not included.

Detailed Explanation: Recognize quadratic relations from tables

Check whether the input values, or $x$ -values, are equally spaced. Then compare the changes in the output values, or $y$ -values.

Consider this table:

$x$	$y$
$0$	$2$
$1$	$5$
$2$	$10$
$3$	$17$
$4$	$26$

Step 1: Check the spacing of the $x$ -values

The $x$ -values increase by $1$ each time:

1,\ 1,\ 1,\ 1

So, we can compare differences in the $y$ -values.

Step 2: Find the first differences

Subtract each $y$ -value from the next one:

5-2=3

10-5=5

17-10=7

26-17=9

The first differences are:

3,\ 5,\ 7,\ 9

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

Step 3: Find the second differences

Now find the differences between the first differences:

5-3=2

7-5=2

9-7=2

The second differences are:

2,\ 2,\ 2

They are equal and nonzero, so the table represents a quadratic relation.

A quadratic relation can be written in the form

y=ax^2+bx+c.

Since the $x$ -values increase by $1$ , the second difference is $2a$ . Here the second difference is $2$ , so $a=1$ . In fact, this table matches

y=x^2+2x+2.

Therefore, equal nonzero second differences show that the relation is quadratic.

Learn by doing: Recognize quadratic relations from tables

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First and Second Differences - Completed Table to Function Type

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