Exercise 9 Page 243

The graph of a linear function is a straight, non-vertical line. To determine whether the relation in the table is a linear function, let's first confirm that the relation is a function.

Words

Since each input $x$ has only one output, the relation in the table is a function. Let's look for a pattern of change in the given table.

Change in $x$	$x$	$y$	Change in $y$
	$0$	$\text{-}3$
$+1$	$1$	$2$	$+5$
$+1$	$2$	$7$	$+5$
$+1$	$3$	$12$	$+5$

We can see that every time the $x\text{-}$values increase by $1,$ the $y\text{-}$values increase by $5.$ This might lead us to believe that the relationship is just $5$ times $x.$ However, by testing the values of $x$ with this relationship, we can see that the resulting values of $y$ are not equal to the given values. Assuming $y$ is equal to $5$ times $x,$ the values of $y$ are $3$ more than we need in each case. Therefore, we can say that the value of $y$ is equal to $5$ times $x$ minus 3.

Equation

To describe the relation with an equation, let's translate our verbal description into algebraic terms.

Graph

Now, let's graph the points. By doing so, we will also be able to determine if the relationship is a linear function.

The graph of a linear function is a straight, non-vertical line. Now that we have graphed the given points, we can see that they are able to be connected with a straight line. The relationship is therefore linear.