Quadratic vertex is one of 4 quadratics formulas in the algebra section of this library, and it is used at high school level.
Why quadratic vertex works
The quadratic formula gives roots at -b over 2a plus and minus the same amount. Their average is therefore -b over 2a, dead centre between them. A parabola is symmetric, so its turning point sits on that centre line, and the height there comes from substituting the value back into the function.
What each symbol means
$a,b$ are coefficients of $f(x)=ax^2+bx+c$.
Quadratic vertex: when it holds
$a\ne0$; the vertex is a minimum if $a>0$ and a maximum if $a<0$.
When it stops applying
It locates the turning point of a parabola that opens up or down. For a sideways parabola written x = ay squared + by + c, the same expression returns the y coordinate of the vertex instead, and reading it as an x coordinate puts the point in the wrong place.
Quadratic vertex: a worked example
For $f(x)=2x^2-8x+1$, $x_v=2$ and $y_v=-7$.
The mistake to avoid
What people do: Dropping the minus sign and writing x = b/(2a), giving 3 instead of -3 for f(x) = x squared + 6x + 5.
Why it goes wrong: That reflects the vertex to the wrong side of the axis. The correct turning point is (-3,-4), while the sign error reports a height of 32 at x = 3.
Do this instead: Write the minus sign as part of the formula, not as something to add later, and sanity-check that the vertex sits between the two roots.
Quadratic vertex: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b$ are coefficients of $f(x)=ax^2+bx+c$.
- Check the conditions before substituting. $a\ne0$; the vertex is a minimum if $a>0$ and a maximum if $a<0$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Quadratics
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where quadratic vertex comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Quadratic Formula Explained — the lesson behind this formula: use the formula and interpret its discriminant.
- Quadratic Formula Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about quadratic vertex
Can I get the height without substituting back?
Yes, the vertex height also equals c minus b squared over 4a. For f(x) = 2x squared - 8x + 1 that is 1 minus 64/8, which is -7, matching the substitution.
Is the vertex always the maximum or minimum of the whole function?
For a parabola, yes. When a is positive the arms rise forever so the vertex is the lowest point, and when a is negative it is the highest point on the entire graph.
How does the vertex relate to the axis of symmetry?
The axis of symmetry is the vertical line drawn through the vertex, written x equals the vertex value. Folding the parabola along that line matches the two arms exactly.
Why does the vertex sit halfway between the two roots?
Because the roots are placed symmetrically around -b over 2a by the plus-or-minus in the quadratic formula. Averaging two numbers that are equally far from a centre returns that centre.