Which of the following is the graph of the equation y = |x| for all

This question can be solved in 10 seconds if this property is known:

V-shaped graph

The graph of a linear absolute value function is V-shaped.

The V-shape can be upright, upside down, or sideways.

There is only one V-shaped graph in this group.

Answer B

Plot points in two phases ("pieces")

If the V-shape is not at your fingertips, then you can plot points, but how absolute value behaves changes "regular" linear graphing.

Linear absolute value functions are written in two pieces, where y = f(x):

\(f(x) = x\), for \(x\geq{0}\)

\(f(x) = -x\), for \(x < 0\)

The first part means you must plot some x-values that are 0 and positive.

The second part means you must choose some x-values that are negative (\(x < 0\)).
[(0,0) can be included with negative x, where (-0 = 0), but that is a long story.]

A note of caution: it might seem perfectly reasonable to plot
\(y = -x\), so one point is \((5, -5)\)

That is arithmetically true, but
It is not consistent with the absolute value function where
-- the variable inside (here, \(x\)), must be considered in both +/- cases (absolute value function makes a negative value positive or non-negative).

Plot three points for both cases

1) \(y = f(x) = x\)

\((0,0), (1, 1), (2, 2)\)

2) \(y = f(x) = -x\): choose points that make the value inside the brackets negative (i.e. \(f(x)\) where \(x < 0\))

\((0,0), (-1, 1), (-2,2)\)

Now there are two straight lines in a V-shape.

Answer B

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