Question

How do you find the vertex, and tell whether the graph `y = 14 - 7/5 abs(x - 7)` is wider or narrower than `y=absx`?

Answer 1

Compare this equation to the basic (or 'parent') equation: `y=abs(x)`

The basic (parent) function is `f(x)=abs(x)`.

The question asks about `y=14-7/5f(x-7)`

This can be rewritten: `(y-14)=-7/5f(x-7)`

Vertex
If we replace `y` by `y-14` and

we replace `x` by `x-7`, then we

translate the graph `+14` in the `y` direction (up 14)

and `+7` in the `x` direction (7 to the right).

So the new vertex is at `(7, 14)`

(Note)
The vertex of `y=absx` is the point where we get `0=abs0`.
In `(y-14)=-7/5 abs(x-7)`, where do we get `0=abs0`? At #(7,14)

Wider or Narrower
Multiplying by a negative reflects the graph across the `x` axis.
(It makes `+` y's negative and vice versa.)

Multiplying the function by a number bigger than `1` (a number with greater absolute value) stretches the graph vertically, making it narrower. (or "taller")