Absolute Value Function

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Absolute Value Function: meaning, method and solved example (Algebra 1) – Maths PDF

In short: Absolute value function is the rule f(x) = |x| and its transformations, whose graph is a V that turns at a corner called the vertex.

Algebra 1Functions and GraphsF-IF.C.7b

What is Absolute Value Function?

The absolute value function \(f(x) = |x|\) measures how far each input sits from zero, so every output is zero or positive. Negative inputs get flipped to the positive side, which bends the graph into a sharp V instead of a smooth curve.

The general version is \(f(x) = a|x – h| + k\). The vertex is \((h, k)\), the line \(x = h\) is the mirror, \(a\) stretches the V and flips it upside down when negative. Unlike a parabola, each arm is perfectly straight.

How to spot it

  • An absolute value function has bars around an expression with x inside, such as \(|x – 2| – 1\).
  • A graph is a V or an upside-down V with straight arms and a pointed vertex.
  • A question asks for a distance from a target value, like how far a part’s length may stray.
  • A shift, stretch or reflection of the basic V is described.

The method

  1. Read h and k from \(a|x – h| + k\): the vertex is \((h, k)\), and a bar with \(x + 3\) means \(h = -3\).
  2. Use the sign of a: positive opens up, negative opens down.
  3. Plot the vertex, then step one unit right and left and move up or down by |a| per unit.
  4. Draw two straight arms from the vertex; find zeros by solving \(a|x – h| + k = 0\).
vertex (2, −1)(1, 0)(3, 0)f(x) = |x − 2| − 1straight arms, slope ±1
f(x) = |x − 2| − 1 is a V with its vertex at (2, −1).

Absolute Value Function: solved example

Problem

Describe \(g(x) = -2|x + 3| + 4\): vertex, direction, and where it crosses the x-axis.

Solution

Writing \(x + 3 = x – (-3)\) gives \(h = -3\), and \(k = 4\), so the vertex is \((-3, 4)\). Since \(a = -2\) is negative, the V opens down and is narrower than the basic one.

For zeros: \(-2|x + 3| + 4 = 0\) gives \(|x + 3| = 2\), so \(x + 3 = 2\) or \(x + 3 = -2\). The graph crosses at \(x = -1\) and \(x = -5\).

Try it

Question 1

Name the vertex of \(f(x) = |x – 5| + 2\).

Show the answer

\(h = 5\) and \(k = 2\), so \((5, 2)\).

Question 2

Evaluate \(f(x) = 3|x| – 4\) at \(x = -2\).

Show the answer

\(3 \cdot 2 – 4 = 2\).

Absolute Value Function vs. Quadratic Function

Both graphs have a turning point and a mirror line, which causes mix-ups. Yet this graph is made of two straight rays meeting at a sharp corner, while the quadratic is one smooth curve with no corner. The bars in the rule are the giveaway.

Quadratic Function

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