## mathmath333 one year ago Prove

1. mathmath333

\large \color{black}{\begin{align} & \normalsize \text{Prove the area of the region enclosed by the graph }\hspace{.33em}\\~\\ & y=-|x\pm k|+5,y=0\hspace{.33em}\\~\\ & \normalsize \text{ is constant for} \ \ k,\ k\in \mathbb{R}\ \hspace{.33em}\\~\\ \end{align}}

2. ganeshie8

I'm finding it hard to visualize... could you show the area in graph ?

3. mathmath333
4. ganeshie8

Ahh that should be easy

5. ganeshie8

First of all, we acknowledge that the the absolute value gives the distance between two points on number line: |dw:1434119749216:dw|

6. perl

You can prove it using calculus. The integral of that is constant.

7. ganeshie8

In light that of above thing, $$|x-k|$$ represents the distance between $$x$$ and $$k$$.

8. ganeshie8

Set the given function equal to $$0$$ and solve $$x$$ intercepts : $-|x-k|+5=0 \implies |x-k|=5$

9. mathmath333

x=5+k.-5+k

10. ganeshie8

Yes, subtract them to get the base of triangle

11. mathmath333

base->10

12. ganeshie8

|dw:1434120100063:dw|

13. mathmath333

ok thnx