#absolute values

Written by @shinyopals on LJ

Tentoo (T)

It turns out, the Doctor in blue on the beach in Journey's End wasn't actually a metacrisis. He was, in fact, the Fourteenth Doctor, back in time to be with Rose again.

The Sums of Things

Inverse Reflections

Absolute Values

Unpredictable Vectors

Identity Element

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In Chapter 1.2 we looked at several basic mathematical characteristics:

reflexive := (a = a)

cumulative := (a + b + c) = (b + a + c)

associative := ( (a + b ) + c)= ( (c+ b ) + a)

distributive := c * (a + b) = ca + cb

identity := (a * 1 = a), (a + 0 = a)

inverse := (a * 1 / a = 1), (a - a = 0)

quotient := (a / b) = (a * (b ^ -1)), where b not equal to 0.  

transitive := if a < b and c < d, then a + c < b + d

In this chapter, we explore the function absolute x, “abs (x)”, or written mathematically ” | x | ”, using equalities first, then looking at inequalities.

reflexive := abs (-a) = a, where a >= 0

cumulative := abs (a + b) <= abs (a) + abs (b), where a >= b

distributive := abs (ab) = abs (a) * abs (b)

identity := abs (a) = sqr (a^2)

quotient := abs (a / b) = abs (a) / abs (b), where b != 0

-1 * abs (x) <= x <= abs (x)

For inequalities, we will look at sets of where (x) is an absolute value that is less than or greater than the value of (a).

A1 = [ x: abs (x) < a, -a < x < a ]

A2 = [ x: abs (x) > a, -a > x > a ]

To graph inequalities on a line using arrows “ < > “, parentheses “ ( ) “ and brackets. “ [ ] “.   I use the vertical bar “ | “ to indicate 0.

For example: abs (x) <= a

-a [ ===== | =====  ] a

OR

For example: abs (x) < a

-a ( ===== | ===== ) a

Area’s of Inequalities

When you look at the equation (y <= ax + b), this is no longer a line, but an area.

XY = [ xy: y <= ax + b ]

XY is the set of points (x, y) where ax + b >= y.

if a = 1 and b = 1

x >= y

when y = 1, x can be 1 ... n when y = 2, x can be 2 ... n when y = 3, x can be 3 ... n when y = m, x can be m ... n

and so on.

|5x + 8| ≤ 35–35 ≤ 5x + 8 ≤ 35–43 ≤ 5x ≤ 27–43/5 ≤ x ≤ 27/5

[Link to the question]

When I went to the site, my jaw dropped because I ask like..the SAME question in my book:

I know I’m a pretty good question writer but damn.

Anyway, the important thing to know here is that absolute value graphs bounce off the x-axis. So your job is to look at the equations and figure out which one could be created if you could “unfold” the graph given over the x-axis. In the case of the QOTD, the answer is y = |x^2 – 1|, because the graph of y = x^2 – 1 (without absolute value) looks like this:

Bounce the stuff under the x-axis above it, and you get the graph in the question: