Absolute Value Transformations

The absolute value parent is a V with its vertex at the origin. One general form, , carries every translation, stretch and reflection this V can undergo — and reading off its vertex and shape takes only a glance at , and .

Concept The general form

Vertex — always at

|a| — vertical stretch (

|a| &gt; 1

) or compression (

0 &lt; |a| &lt; 1

Sign of a — negative reflects the V in the x-axis, so it opens downward.

As with every parent function, the sign inside the bracket is reversed: $h > 0$ moves the vertex right, and $h < 0$ moves it left. Outside the bracket, behaves as expected: positive lifts, negative lowers.

Example Reading off the vertex

Find the vertex of .

Matching against

gives

and

The vertex of any member of this family is

⟹ vertex (4, 2)

Example Reading off the shape

Describe compared with the parent .

Here

, so

|a| = 3 &gt; 1

: the V is stretched vertically, narrower than the parent.

Since

a &lt; 0

, the graph is reflected in the x-axis and opens downward.

⟹ a narrow V, opening downward, vertex still at the origin

Concept Solving absolute value equations

Because measures distance from zero, the equation (for $c > 0$ ) always has two solutions: the two points units from zero.

splits into: expression

, or expression

Graphically, this is exactly where the horizontal line crosses both arms of the V.

Example Solving an equation

Solve .

Case 1:

Case 2:

Check: both 5 and −9 are exactly 7 units from −2.

⟹ x = 5 or x = −9

Example Building the equation from a description

Write the function whose V has vertex and is twice as narrow as .

A negative

becomes

inside the bracket:

, added outside to lower the vertex.

Twice as narrow means

⟹ f(x) = 2|x + 1| − 3

Checking: , matching the required vertex.

Summary

In f(x) = a|x − h| + k, the vertex is always (h, k).
|a| stretches (|a| > 1) or compresses (0 < |a| < 1) the V; a negative a reflects it downward.
|expression| = c splits into two linear equations, expression = c and expression = −c.
The same reversed-sign rule for h applies here as for every other parent function.