Algebra
Quadratic equations and functions
Vertex form of a quadratic
20 practice questions
2 video lessons
Theory + worked examples
Theory
Vertex form shows the vertex directly:
\[y=a(x-h)^2+k,\]
with vertex \((h,k)\), axis of symmetry \(x=h\), and \(a\) controlling direction and width.
Watch the sign of \(h\): \((x+3)\) means \(h=-3\).
Vertex \((2,1)\) read straight from the form.
Vertex form.
Vertex form:
\[y=a(x-h)^2+k\]
\((h,k)\) is the vertex; \(x=h\) is the axis.
How to use vertex form
- Read \(h\) (opposite sign inside).
- Read \(k\) (the constant).
- The vertex is \((h,k)\).
- Use \(a\) for direction and width.
Example 1 — Read the vertex
Give the vertex of \(y=(x-2)^2+1\).
Solution
Read \(h\) and \(k\).
| \(\text{vertex}\) | \(=\) | \((2,1)\) |
Example 2 — Watch the sign
Give the vertex of \(y=(x+3)^2-5\).
Solution
\((x+3)\) means \(h=-3\).
| \(\text{vertex}\) | \(=\) | \((-3,-5)\) |
Example 3 — Direction
Does \(y=-2(x-1)^2+4\) open up or down?
Solution
\(a=-2<0\), so down; vertex is a maximum.
| \(a<0\) | \(\Rightarrow\) | \(\text{opens down}\) |
Example 4 — Axis of symmetry
State the axis of symmetry of \(y=(x-2)^2+1\).
Solution
The axis is \(x=h\).
| \(x\) | \(=\) | \(2\) |
Common pitfalls
\((x-h)\) gives \(h\) with the opposite sign.
\(k\) shifts vertically; \(h\) horizontally.
\(a<0\) opens down.
Frequently asked questions
What is vertex form?
\(y=a(x-h)^2+k\), showing the vertex \((h,k)\).
What is the vertex of \(y=(x-4)^2+2\)?
\((4,2)\).
What is the vertex of \(y=(x+1)^2-3\)?
\((-1,-3)\).
What is the axis of symmetry in vertex form?
\(x=h\).
← Previous subtopic
Quadratic functions and graphs
Next subtopic →
Transformations of quadratic functions
More in Quadratic equations and functions