Exercise 63 Page 162

The best way to understand how different transformations change graphs is by graphing a few examples and comparing the differences. Ultimately, we want to know how $a$ affects the vertex of the absolute value function. When $a=1$ the vertex is $(h,k).$ First let's test a couple of functions, keeping $h=1$ and $k=1.$ We can see how the vertex changes from a function where $a=1$ — where we already know that the vertex is $(1,1)$ — to functions with $a=2$ and $a=\frac{1}{4}.$ This will give us an idea of how $a$ changes the vertex.

As shown above, when $a=2$ the vertex becomes $(\frac{1}{2},1).$

When $a=\frac{1}{4},$ the vertex becomes $(4,1).$ Let's compare these functions and vertices.

$a$	$f(x)=|ax-1|+1$	Vertex
$1$	$f(x)=|x-1|+1$	$(1,1)$
$2$	$f(x)=|2x-1|+1$	$\left(\frac{1}{2},1\right)$
$\frac{1}{4}$	$f(x)=\left|\frac{1}{4}x-1\right|+1$	$(4,1)$

We can see that we can create the $x\text{-}$coordinate of the vertex by dividing $h$ or $k$ by $a.$ We can graph one more function with different values for $h$ and $k$ to confirm this. Let's graph the function $f(x)=|2x-4|+3.$

The vertex of this function is $(2,3).$ Since $4÷2=2,$ we can conclude that we are dividing $h$ by $a.$ Now we can write a general form for the vertex.