Exercise 41 Page 259

We have been given the following function rules. We are asked to make a table of values and graph these functions in order to determine the change of the $y\text{-}$values when we double the $x\text{-}$values. Let's examine each function rule separately.

$y=2x$

Let's make a table of values for $y=2x$ and graph it!

$x$	$2x$	$y=2x$	$(x,y)$
$\text{-}1$	$2\cdot (\text{-}1)$	$\text{-}2$	$(\text{-}1,\text{-}2)$
$0$	$2\cdot 0$	$0$	$(0,0)$
$1$	$2\cdot 1$	$2$	$(1,2)$
$2$	$2\cdot 2$	$4$	$(2,4)$

Let's graph these points on a coordinate plane.

Now, let's double the $x\text{-}$values.

$x$	$2x$	$y=2x$	$(x,y)$
$\text{-}2$	$2\cdot (\text{-}2)$	$\text{-}4$	$(\text{-}2,\text{-}4)$
$0$	$2\cdot 0$	$0$	$(0,0)$
$2$	$2\cdot 2$	$4$	$(2,4)$
$4$	$2\cdot 4$	$8$	$(4,8)$

Let's graph our new values.

As we can see, when we double the $x\text{-}$values of $y=2x,$ the $y\text{-}$values are also doubled but the graph remains the same.

$y=2x^2$

Now, let's apply the same process for $y=2x^2.$

$x$	$2x^2$	$y=2x^2$	$(x,y)$
$\text{-}2$	$2\cdot (\text{-}2{)}^2$	$8$	$(\text{-}2,8)$
$\text{-}1$	$2\cdot (\text{-}1{)}^2$	$2$	$(\text{-}1,2)$
$0$	$2\cdot 0^2$	$0$	$(0,0)$
$1$	$2\cdot 1^2$	$2$	$(1,2)$
$2$	$2\cdot 2^2$	$8$	$(2,8)$

The corresponding graph for $y=2x$ is the following.

Now let's see how the table changes when we double the $x\text{-}$values.

$x$	$2x^2$	$y=2x^2$	$(x,y)$
$\text{-}4$	$2\cdot (\text{-}4{)}^2$	$32$	$(\text{-}4,32)$
$\text{-}2$	$2\cdot (\text{-}2{)}^2$	$8$	$(\text{-}2,8)$
$0$	$2\cdot 0^2$	$0$	$(0,0)$
$2$	$2\cdot 2^2$	$8$	$(2,8)$
$4$	$2\cdot 4^2$	$32$	$(4,32)$

Let's graph the new points.

In this case, when we double the $x\text{-}$values, the $y\text{-}$values quadruple but the graph again remains the same.