Vertex Form Calculator
Convert between standard form and vertex form of quadratic equations
Convert Quadratic Equations
Standard Form: y = ax² + bx + c
Cannot be zero
Conversion Results
Standard Form
Vertex Form
Parabola Properties
Step-by-step Conversion
Example Calculations
Example 1: Standard to Vertex
Given: y = 2x² + 8x + 3
Solution:
h = -8/(2×2) = -8/4 = -2
k = 3 - 8²/(4×2) = 3 - 64/8 = 3 - 8 = -5
Vertex Form: y = 2(x + 2)² - 5
Vertex: (-2, -5)
Example 2: Vertex to Standard
Given: y = -3(x - 1)² + 4
Solution:
(x - 1)² = x² - 2x + 1
y = -3(x² - 2x + 1) + 4
y = -3x² + 6x - 3 + 4
Standard Form: y = -3x² + 6x + 1
Example 3: Perfect Square
Given: y = x² - 6x + 9
Analysis: This is a perfect square trinomial
h = -(-6)/(2×1) = 6/2 = 3
k = 9 - (-6)²/(4×1) = 9 - 36/4 = 9 - 9 = 0
Vertex Form: y = (x - 3)²
Vertex: (3, 0) - touches x-axis at one point
Conversion Formulas
Standard to Vertex
Vertex to Standard
Key Properties
Parabola Types
a > 0: Opens upward
Vertex is minimum point
a < 0: Opens downward
Vertex is maximum point
|a| > 1: Narrow parabola
Steeper curve
|a| < 1: Wide parabola
Flatter curve
Real-World Applications
Physics
Projectile motion, trajectory calculations
Economics
Profit maximization, cost minimization
Engineering
Bridge design, satellite dishes
Optics
Parabolic reflectors, telescopes
Understanding Vertex Form
What is Vertex Form?
The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) represents the vertex of the parabola. This form makes it easy to identify the vertex and understand the parabola's key properties.
Why Use Vertex Form?
- •Immediately shows the vertex coordinates
- •Makes transformations obvious
- •Simplifies graphing the parabola
- •Useful for optimization problems
Parameter Meanings
a: Vertical stretch and direction
Same as in standard form
h: Horizontal shift
Moves parabola left/right
k: Vertical shift
Moves parabola up/down
Remember: In y = a(x - h)² + k, if h is positive, the parabola moves RIGHT, not left. The minus sign in the formula can be confusing!