# 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?

Apr 1, 2015

Compare this equation to the basic (or 'parent') equation: $y = \left\mid x \right\mid$

The basic (parent) function is $f \left(x\right) = \left\mid x \right\mid$.

The question asks about $y = 14 - \frac{7}{5} f \left(x - 7\right)$

This can be rewritten: $\left(y - 14\right) = - \frac{7}{5} f \left(x - 7\right)$

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 $\left(7 , 14\right)$

(Note)
The vertex of $y = \left\mid x \right\mid$ is the point where we get $0 = \left\mid 0 \right\mid$.
In $\left(y - 14\right) = - \frac{7}{5} \left\mid x - 7 \right\mid$, where do we get $0 = \left\mid 0 \right\mid$? 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")