Exercise 1 Page 42

Next, we will look for the slope and the $y\text{-}$intercept in the given graph.

We can see that the slope of this graph is $2$ and its $y\text{-}$intercept is $1.$

Function	Slope	$y\text{-}$intercept
$f(x)$	$2$	$\text{-}4$
$g(x)$	$2$	$1$

The function $g(x)$ has the same slope as $f(x),$ but their $y\text{-}$intercepts are different. To obtain $g$ we must translate the graph of $f$ $5$ units up.

Let's look for the slope and the $y\text{-}$intercept in the given graph.

We can see that the slope of this graph is $1$ and its $y\text{-}$intercept is $\text{-}2.$ This means its equation is given by $g(x)=x-2.$ Using this equation, we can set a relation between $f(x)$ and $g(x).$ To obtain $g$ we must shrink the graph of $f$ vertically by a factor of $\frac{1}{2}.$

Now, we look for the slope and the $y\text{-}$intercept in the given graph.

We can see that the $y\text{-}$intercept of of this graph is $\text{-}4,$ but we cannot determine the slope of it. However we see that it must be greater than $2,$ since it is more vertical than the graph of $f(x).$ This means that to obtain $g$ we must shrink the graph of $f$ horizontally.

Let's also study the graph of the function $g(x).$

In the graph above, we see that the rise is $4$ when the run is $2.$ This implies that the slope is $m=\frac{2}{1}=2.$ Additionally, the $y\text{-}$intercept is $\text{-}12.$ We can now compare this data with that from the parent function.

Function	Slope	$y\text{-}$intercept
$f(x)$	$2$	$\text{-}4$
$g(x)$	$2$	$\text{-}12$

The function $g(x)$ has the same slope as $f(x),$ but their $y\text{-}$intercepts are different. To obtain $g$ we must translate the graph of $f$ $8$ units down.

Now, let's take a look at the graph of the function $g(x).$

As we can see, for every unit we move to the right, points on the line move two units down. This means that the slope is $\text{-}2.$ Additionally, the $y\text{-}$intercept is $4.$ By using the slope-intercept form we can find the equation of this line. Additionally, we can state the following relation between the functions $f(x)$ and $g(x).$ To obtain $g$ we must reflect the graph of $f$ across the $x\text{-}$axis.

Next, we will take a look at the graph of the function $g(x).$

As we can see, for every unit we move to the right, points on the line move two units down. This means that the slope is $\text{-}2.$ Additionally, the $y\text{-}$intercept is $\text{-}4.$ By using the slope-intercept form we find the equation of this line. We can state the following relation between the functions $f(x)$ and $g(x).$ Therefore, to obtain $g$ we must reflect the graph of $f$ across the $y\text{-}$axis.