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- Converts a quadratic from standard form ax² + bx + c to vertex form a(x − h)² + k.
- Also converts vertex form back to standard form.
- Gives the vertex, axis of symmetry, roots, y-intercept, focus and directrix.
- Shows the completing-the-square formulas step by step.
How to use the vertex form calculator
Choose the direction
Standard to vertex form, or vertex to standard form.
Enter the coefficients
a, b and c for standard form, or a, h and k for vertex form.
Read the converted equation
Both forms of the parabola are shown.
Use the extras
The vertex, roots, focus and directrix are listed for graphing.
How the conversion works
Standard form to vertex form
h = −b ÷ 2a
k = c − b² ÷ 4a
y = a(x − h)² + k
This is completing the square written as two quick formulas.
Vertex form to standard form, focus and directrix
b = −2ah, c = ah² + k
Focus = (h, k + 1 ÷ 4a), Directrix: y = k − 1 ÷ 4a
Example 1: y = 2x² − 8x + 3
a = 2, b = −8, c = 3.
- h 8 ÷ 4 = 2
- k 3 − 64 ÷ 8 = −5
- Vertex form y = 2(x − 2)² − 5
- Roots x ≈ 3.581139 and 0.418861
- Focus and directrix (2, −4.875), y = −5.125
Example 2: other parabolas
More conversions.
- x² − 4x + 4 (x − 2)², touches the x-axis at 2
- x² + 2x + 5 (x + 1)² + 4, no real roots
- −x² + 9 vertex (0, 9), opens downward, roots ±3
Where the vertex form helps
Graphing parabolas
The vertex form shows the turning point and shift at a glance.
Max and min problems
Find the highest or lowest value of a quadratic, such as maximum profit.
Projectile paths
Find the peak height of an object in flight.
Algebra homework
Check completing-the-square answers.
Find exact or complex roots with the quadratic formula calculator, the slope of a line with the slope calculator, and real flight paths with the projectile motion calculator.
Vertex form calculator FAQ
What is vertex form?
A way of writing a quadratic as y = a(x − h)² + k, where (h, k) is the vertex of the parabola.
How do I convert to vertex form?
Find h = −b ÷ 2a and k = c − b² ÷ 4a, then write y = a(x − h)² + k.
How do I find the vertex of a parabola?
From standard form, x = −b ÷ 2a, then substitute to find y. The calculator does both.
Does a change in vertex form?
No. The same a appears in both forms and controls how wide the parabola is and which way it opens.
What are the focus and directrix?
Every point on a parabola is the same distance from the focus point and the directrix line. They sit 1 ÷ 4a above and below the vertex.
What if there are no roots?
The parabola does not cross the x-axis, so there are no real roots. The quadratic formula calculator shows the complex ones.