Vertex Calculator and Vertex Form Calculator
This vertex calculator finds the turning point of a parabola from a, b and c, writes it out in vertex form, draws the curve, and — the part that matters most — keeps the value and where it happens in two separate sentences.
Type 2, −8, 3 and you get (2, −5) and y = 2(x − 2)² − 5: a minimum value of −5, reached at x = 2.
What the Vertex Is, and What It Is Not
The vertex is where a parabola turns — the bottom of the U if it opens upwards, the top if it opens down. It is a point, so it is two numbers.
And that is where the trouble starts, because the two numbers answer different questions and are not interchangeable. For a vertex at (2.5, −0.5):
| −0.5 | the minimum value | how low the curve gets |
| 2.5 | where it happens | the x that produces it |
Reporting "the minimum is 2.5" swaps the input for the output. It is an easy slip to make in a hurry and it appears in published lessons often enough that you should not assume a printed answer has it the right way round. Write the two on separate lines, in words, and the problem disappears.
Minimum or Maximum: the Sign of a Decides
Nothing else does. a positive and the arms point up, so the turning point is the lowest the curve ever goes. a negative and the arms point down, making it the highest.
y = 2x² − 8x + 3 has a minimum at (2, −5).
y = −x² + 4x − 1 has a maximum at (2, 3).
This is the half of the answer most often left off. A question asking for the greatest height a ball reaches, or the price that maximises revenue, wants a number and a word — "the maximum is 3, at x = 2" — not a bare pair of coordinates for the marker to interpret.
Why It Sits at −b Over 2a
A parabola is symmetric. Fold it down the middle and the two halves land on each other exactly, so the turning point must sit halfway between the two places it crosses the x-axis.
Those two crossings add up to −b/a. Halve that and you have −b/2a. That is the whole derivation, and it explains something the formula alone does not: it keeps working when the curve never reaches the axis at all. There are no crossings to average, but the symmetry is still there, so the midpoint still exists.
The minus is part of the formula and is the piece people drop. With b = −8, −b is +8, so x = 8 ÷ 4 = 2. Write that substitution out with the signs showing before simplifying anything; it costs one line.
Then put your x back into the equation to get y. Not into a second formula — into the curve itself. Fewer things to remember, and fewer to get wrong.
Vertex Form: the Same Curve, Written Differently
Once you have the vertex there is nothing left to calculate. Vertex form is just those two numbers put into a shape that displays them:
y = a(x − h)² + k
So y = 2x² − 8x + 3, whose vertex is (2, −5), becomes y = 2(x − 2)² − 5. The a is unchanged — converting to vertex form never alters how wide the curve is or which way it opens, only where the turning point is written down.
Four things go wrong when people write it by hand, and all four are about presentation rather than arithmetic:
| The bracket flips | h = −3 gives (x + 3)², not (x − 3)² |
| A negative k | write − 5, never + −5 |
| a = 1 or −1 | (x − 2)² and −(x − 2)², with no 1 in sight |
| h or k is zero | h = 0 gives x²; k = 0 is simply left off |
This page follows all four, so what it prints is what a marker expects to see rather than a mechanical substitution. And if you would rather arrive at the same equation the long way, completing the square gets there step by step and shows why the bracket comes out the way it does.
Reading It Off a Bracket — Watch the Sign
When a quadratic is written as a squared bracket, the number you can see is the opposite of where the curve turns.
y = (x + 3)² − 4 turns at x = −3, not +3.
The reason is worth holding onto rather than memorising: a square is never negative, so the whole expression is smallest exactly when the bracket is zero — and x + 3 is zero at −3. That is why the warning above matters more in vertex form than anywhere else: the sign you can see is never the sign of the answer.
What the Graph Shows You
The picture on this page marks four different things, and keeping them apart is most of the skill:
| Filled dot | the vertex |
| Dashed line | the axis of symmetry, x = h |
| Hollow dots | where it crosses the x-axis, if it does |
| y-axis crossing | the y-intercept, which is simply c |
The crossings are a different question from the vertex, and the quadratic formula is what answers it. A curve can have a perfectly ordinary vertex and never touch the x-axis at all: y = x² + 2x + 5 turns at (−1, 4) and stays above the axis forever, because its lowest point is already 4 units up.
Checking It by Symmetry
Step one unit either side of your x and work out both y values.
For y = 2x² − 8x + 3 with a vertex at x = 2: at x = 1 the value is −3, and at x = 3 it is also −3.
If those two came out different, the x is wrong — because a turning point is the one place a parabola mirrors itself about, and nowhere else on the curve has that property. Both should also be worse than the value you found: further up for a minimum, further down for a maximum. Fifteen seconds, and it catches a dropped minus sign, which is the error this topic produces more than any other.
Using the Vertex Calculator
Read a, b and c straight off y = ax² + bx + c and type them with their signs. Nothing needs moving to one side — a turning point belongs to the whole curve, not to a value it is set equal to, so there is no rearranging to do first.
Leave b or c alone if that term is missing. Fractions like 3/4 work, decimals work, and the answer stays exact: y = x² − 5x + 6 comes back as (5/2, −1/4) rather than (2.5, −0.25). That is worth having, because a vertex lands on a half or a quarter far more often than it lands on a whole number.
Open the working and you get the substitution with the signs still raw, the value found by putting x back in, the graph, the two numbers said apart, and the symmetry check with your own numbers in it.
Vertex Calculator FAQ
How do you find the vertex of a parabola?
Is the vertex a minimum or a maximum?
What is the minimum value of a quadratic?
Why is the vertex at minus b over 2a?
Where does y equals x plus 3 all squared turn?
What is vertex form, and how do you get it?
How do you check the vertex?
Minus b over 2a, then back into the curve — say the value and the place in two separate breaths, and mind the bracket when you write it out.