What does this graph look like?

Algebra Level 3

$\large y = \min ( \, \color{#20A900}{ \left|x \right|} \, , \, \color{#D61F06} {1 + \left| \left| x \right| + x \right|} \, , \, \color{#3D99F6}{2 + \left| \left|x \right| - x \right|} \, )$

On the interval $[ -2, 4]$ , what does the graph of $y$ look like?

M W Resistor symbol Square root symbol

2 solutions

Eli Ross Staff
Oct 9, 2015

One way to graph this function is to graph all 3 equations. It is useful to note that $\color{#D61F06}{\left||x|+x\right|=2x}$ for $x\ge 0$ and $\color{#D61F06}{0}$ for $x<0,$ while $\color{#3D99F6}{\left||x|-x\right|=0}$ for $x \ge 0$ and $\color{#3D99F6}{2x}$ for $x<0.$

Then, we can "trace" over the minimum of the three graphs:

Alas, we have a square root symbol!

Haha. Yea, that's how I came up with the problem.

I wanted to convert it into a single function, but then that became too complicated.

Calvin Lin Staff - 5 years, 8 months ago

I like the Heaviside step function for this reason.

$y = 1 + (|x| - 1)u(x + 1) + (2 - |x|)u(x - 2)$

Okay, well maybe it's not any less complicated...

Andrew Ellinor - 5 years, 8 months ago

Chew-Seong Cheong
Oct 11, 2015

$\begin{aligned} y & = \min (\color{#20A900}{|x|}, \color{#D61F06}{1+||x|+x|}, \color{#3D99F6}{2+||x|-x|}) \\ & = \begin{cases} x < 0 & \Rightarrow \min (\color{#20A900}{|x|}, \color{#D61F06}{1}, \color{#3D99F6}{2+2|x|}) & = \begin{cases} -2 \le x < -1 & \Rightarrow \color{#D61F06}{1} \\ -1 \le x < 0 & \Rightarrow \color{#20A900}{|x|} \end{cases} \\ x \ge 0 & \Rightarrow \min (\color{#20A900}{|x|}, \color{#D61F06}{1+2|x|}, \color{#3D99F6}{2}) & = \begin{cases} 0 \le x < 2 & \Rightarrow \color{#20A900}{|x|} \\ 2 \le x \le 4 & \Rightarrow \color{#3D99F6}{2} \end{cases} \end{cases} \end{aligned}$

The final graph is as follows -- a $\boxed{\text{square-root symbol}}$ .

What does this graph look like?

2 solutions

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