Question

How do you graph `y = -2|x-4|+4`?

Answer 1

Try getting rid of the absolute value operator by rewriting it as a piecewise function.
graph{-2abs(x-4)+4 [-10, 10, -5, 5]}

Explanation:

The inner function of the absolute value operator has a single zero at `x=4`. Thus the output of the absolute value would be linear on ranges `(-infty , 4)` and `(4, infty)`. Hence it would be possible to define an equivalent piecewise function with no absolute values on the two ranges.

`y=-2color(blue)([color(red)(-)color(black)((x-4))])+4=2(x-4)+4=2x-4`,
`x in (-infty , 4)`

`y=-2color(blue)([color(black)((x-4))])+4=-2x+12`,
`x in (4, infty)`

Note that you'll need to reverse the sign of the inner function when removing the absolute value operator in case the expression inside the absolute value yields a negative value.

Plotting the two functions on their respective range gives:

Combining the two rays will give the final v-shaped curve.
graph{-2abs(x-4)+4 [-4, 16, -5, 5]}