Absolute Value Functions
V-shaped graph.
Key idea
V-shaped graph.
In depth
f(x) = |x| reflects the negative side upward, creating a V shape with vertex at the origin. Written piecewise, |x| = x for x ≥ 0 and −x for x < 0.
Worked example
f(x) = |x − 2| + 1 has vertex (2, 1), with slopes ±1.
Formula
Real-world uses
- Modelling distance from a target value, such as tolerance in manufacturing or error in measurements
Common mistake
Expecting a smooth curve; it has a sharp corner at the vertex.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Absolute Value Functions?
V-shaped graph. It is a topic at the advanced level in Algebra.
What is the formula for Absolute Value Functions?
The key formula is y = a|x−h|+k.
Where is Absolute Value Functions used in real life?
Real-world examples: Modelling distance from a target value, such as tolerance in manufacturing or error in measurements.
What should I learn before Absolute Value Functions?
Make sure you are comfortable with Piecewise Functions, the previous topic in Algebra.
What is a common mistake with Absolute Value Functions?
Expecting a smooth curve; it has a sharp corner at the vertex.
How can I practise Absolute Value Functions?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Absolute Value Functions?
Move on to Function Transformations next.
Related advanced topics in Algebra
- Absolute Value Inequalities – Rewrite as a compound inequality.
- Systems with Three Variables – Reduce to two variables then solve in sequence.
- Systems of Inequalities – Solution region is the overlap of half-planes.
- Factoring by Grouping – Group terms in pairs to find a common binomial.
- Sum and Difference of Cubes – Special factoring patterns for cubes.
- Completing the Square – Rewrite a quadratic as a perfect square plus constant.