Chapter 14, Problem 53CCSR

Solution

To find: The vertex of the graph of a given function.

The vertex of the function $f\left(x\right)=a\left|bx-1\right|+c$ is $x=\frac{1}{b}y=k$ .

Given data: The given function is $f\left(x\right)=a\left|bx-1\right|+c$

Method/Formula used:

The function $g\left(x\right)=p\left|x-q\right|+r$ has minimum value at point $x=q,y=r$ that is its vertex.

Calculation:

The given function is $f\left(x\right)=a\left|bx-1\right|+c$ that can be written as,

  $f\left(x\right)=\frac{a}{b}\left|x-\frac{1}{b}\right|+c$

Compare functions $g\left(x\right)$ and $f\left(x\right)$ , $q=\frac{1}{b},r=c$ . Therefore, the vertex of $f\left(x\right)=\frac{a}{b}\left|x-\frac{1}{b}\right|+c$ or $f\left(x\right)=a\left|bx-1\right|+c$ is $x=\frac{1}{b}y=k$ or $\left(\frac{1}{b},k\right)$

Verification:

Consider function $f\left(x\right)=2\left|4x-1\right|+3$ , from the above results the vertex of the function $f\left(x\right)=2\left|4x-1\right|+3$ is $\left(0.25,3\right)$ .

The graph of the function clearly the vertex of the function $f\left(x\right)=2\left|4x-1\right|+3$ is $\left(0.25,3\right)$ .