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

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

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Answer 1 · 2015-04-01T16:20:10

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

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

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

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

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

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

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

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

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

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

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

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

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

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