Algebra
Other functions
Absolute value functions
20 practice questions
2 video lessons
Theory + worked examples
Theory
The absolute value function graphs as a V:
\[y=a|x-h|+k,\]
with vertex \((h,k)\). It opens up if \(a>0\), down if \(a<0\).
The vertex is the turning point of the V.
\(|x-1|+1\) is the V shifted right 1, up 1.
The absolute value function.
The form:
\[y=a|x-h|+k\]
Vertex \((h,k)\); the sign of \(a\) sets the direction.
How to graph an absolute value function
- Find the vertex \((h,k)\).
- Decide the direction from \(a\).
- Plot the vertex and points on each side.
- Draw the V.
Example 1 — The parent
Describe the graph of \(y=|x|\).
Solution
A V with vertex at the origin.
| \(\text{vertex}\) | \(=\) | \((0,0)\) |
Example 2 — Find the vertex
Find the vertex of \(y=|x-1|+1\).
Solution
Read \(h\) and \(k\).
| \(\text{vertex}\) | \(=\) | \((1,1)\) |
Example 3 — Opens down
Does \(y=-|x|+3\) open up or down?
Solution
\(a=-1<0\), so it opens down.
| \(a<0\) | \(\Rightarrow\) | \(\text{opens down}\) |
Example 4 — Evaluate
Find \(y\) when \(x=-2\) for \(y=|x|+1\).
Solution
Take the absolute value.
| \(|-2|+1\) | \(=\) | \(3\) |
Common pitfalls
\((x-h)\) gives \(h\) with the opposite sign.
A negative \(a\) opens the V downward.
Absolute value is never negative.
Frequently asked questions
What shape is an absolute value graph?
A V.
What is the vertex of \(y=|x-2|+3\)?
\((2,3)\).
When does the V open down?
When \(a<0\).
What is the vertex of the parent \(y=|x|\)?
The origin.
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