Exercise 27 Page 158

We are asked to graph $f(x)=3x+4$ and $h(x)=f(6x).$ Also, we need to find the transformations from the graph of $f$ to the graph of $h.$

Graph of $f(x)$

To graph $f(x),$ we will first make a table of values.

$x$	$3x+4$	$f(x)$
$0$	$3(0)+4$	$4$
$6$	$3(6)+4$	$22$
$12$	$3(12)+4$	$40$

Now, we can plot these points and connect them with a straight line to have the graph of $f(x).$

Graph of $h(x)$

Let's recall the definition of a horizontal stretch and shrink. By multiplying the input of a function by a factor $a>0,$ its graph can be horizontally stretched or shrunk by a factor of $\frac{1}{a}$ from the original function. A value of $a$ less than $1$ represents a horizontal stretch of the original function. Conversely, when $a$ is "greater than" $1,$ the function is horizontally shrunk. We can see that $h$ is written in the form $y=f(a\cdot x).$ In this case, $a$ is $6.$ Since $6>1,$ $h(x)$ is a vertical shrink from $f$ by a factor of $\frac{1}{6}.$ Let's take some values for $x,$ then evaluate $f$ in $6x$ to get $h.$

$x$	$f\left(6x\right)$	$h(x)$
$0$	$3(6(0))+4$	$4$
$1$	$3(6(1))+4$	$22$
$2$	$3(6(2))+4$	$40$

Now, we can graph $h$ and $f$ in the same coordinate plane.