how to write absolute value equation into piceswise function?

Question

how do you write this as a piecewise function?
2 |x-2| +5

Answer 1

For this, you have to consider that |x - 2| is always going to be positive, and that x = 2 is the point where everything gets symmetrical (i.e. at x = 1, you have the same value as x = 3, etc.).

If we were to graph this, x = 2 is, again, a pivotal point. Anything greater than x = 2 would mean that we're basically graphing y = 2(x - 2) + 5. Getting this into y-intercept form, we have:

y = 2(x - 2) + 5
y = 2x - 4 + 5
y = 2x + 1

Now, anything less than x = 2 would mean that we're graphing y = 2[-(x - 2)] + 5 - the opposite of the absolute value comes into play here. Simplifying this, we have:

y = 2[-(x - 2)] + 5
y = -2x + 4 + 5
y = -2x + 9

So, piecewise your function becomes:

y = 2x + 1, x >= 2
y = -2x + 9, x < 2

Answer 2

These are not perp lines. What do you want to do with them? explain?