Exercise 1 Page 170

Let's choose the values to support our claim first, then we will rearrange them to support our friend's claim. Keep in mind that these are just one possible answer for this problem.

Our Claim

We want to create a table of values that represents a linear function, which means the rate of change has to be unchanging. From the given numbers, we see that both the top and the bottom tiles increase by $1.$ Since the increase is linear in both rows, let's choose the first $4$ top tiles as our $x\text{-}$values and the first $4$ bottom tiles as the $y\text{-}$values.

$x$	$y$
$\text{-}4$	$1$
$\text{-}3$	$2$
$\text{-}2$	$3$
$\text{-}1$	$4$

When $x$ increases by $1$ the value of $y$ increases by $1.$ Therefore, this is a linear function.

Friend's Claim

Our friend wants to create a table of values that represents a nonlinear function. This means that the rate of change will not be constant. To do this, we can exchange $2$ of the $y\text{-}$values from the table that supported our claim. Let's switch $y=1$ with $y=2.$

$x$	$y$
$\text{-}4$	$2$
$\text{-}3$	$1$
$\text{-}2$	$3$
$\text{-}1$	$4$

Now the rate of change is not constant, so it is a nonlinear function.