Vertex Form Calculator
Completing the square rewrites a quadratic around its vertex: y = 2x² − 8x + 5 becomes y = 2(x − 2)² − 3, with vertex (2, −3).
y = 2(x − 2)² − 3
- Standard form
- y = 2x² − 8x + 5
- Vertex (h, k)
- (2, -3)
- Opens
- upward (minimum at the vertex)
- Minimum value
- -3
- y-intercept
- (0, 5)
Step-by-step solution
- h = −b ÷ 2a = 8 ÷ 4 = 2
- k = c − a·h² = 5 − 2(2)² = -3
- y = 2(x − 2)² − 3
Vertex Form Formula
y = a(x − h)² + k, h = −b/2a, k = c − ah²
vertex (h, k); a > 0 opens up, a < 0 down
Worked Example
The parabola y = 2x² − 8x + 5:
- h = −b ÷ 2a = 8 ÷ 4 = 2
- k = c − a·h² = 5 − 2(2)² = -3
- y = 2(x − 2)² − 3
Vertex (h, k): (2, -3) · Opens: upward (minimum at the vertex) · Minimum value: -3 · y-intercept: (0, 5)
Test-day tip
When an SAT question asks which form “displays the maximum value as a constant”, the answer is vertex form: k is the maximum or minimum, and h is where it occurs.
More Advanced Math Calculators
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Quadratic Formula Calculator
x = [−b ± √(b² − 4ac)] ÷ 2a
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Example: The equation x² − 5x + 6 = 0: Roots: x = 2, x = 3.
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Exponential Growth Calculator
y = a(1 ± r)ᵗ
a initial value, r rate per period, t periods; continuous: y = a·e^(±rt)
Example: 500 growing 8% per period for 5 periods: Value after t periods = 734.6640384.
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Cubic Equation Calculator
ax³ + bx² + cx + d = 0
roots sum to −b/a and multiply to −d/a
Example: The equation x³ − 6x² + 11x − 6 = 0: Roots: x = 3, x = 2, x = 1.
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Simplify Radicals Calculator
√(k²·m) = k√m
take out the largest perfect square (or cube) factor
Example: The square root of 72: Simplified = 6√2.
Frequently asked questions
How do I find the vertex without completing the square?
Use h = −b/2a for the x-coordinate, then substitute h into the equation for k.
What does vertex form tell you that standard form does not?
The vertex (h, k) directly: the maximum or minimum value and where it occurs, which SAT questions about projectiles and profit ask for.
Sources
Last reviewed: . Factors are stored exactly and computed with 50-digit decimal arithmetic. See how we verify conversion factors.
Cite this page
APA
Scientific Converters. (2026, October 15). Vertex Form Calculator. Retrieved October 25, 2026, from https://scientificconverters.com/calculators/vertex-form-calculator/
IEEE
Scientific Converters, "Vertex Form Calculator," scientificconverters.com. Accessed: Oct. 25, 2026. [Online]. Available: https://scientificconverters.com/calculators/vertex-form-calculator/