Skill: 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.