Given a parabola passing through three points (1,2), (2,5), and (3,10), let's use this calculator to quickly find its equation.
When to Use This Tool
- Physics motion modeling: For example, basketball shooting trajectories or projectile motion. Record three measured position points to instantly get the fitted parabola equation, making it easy to predict the landing spot or maximum height.
- Math homework/exam checking: Given any three points on a quadratic function, if you need to write the analytical expression, vertex, and axis of symmetry, use this calculator to get results in a second and compare them with your manual calculations.
- Quadratic models in economics and engineering: For quadratic relationships like cost functions and demand curves, quickly deduce coefficients from three sample points to assist in decision-making.
Principles and Formulas
The standard form of a parabola (quadratic function) is y = ax² + bx + c. Given three non-collinear points (x₁,y₁), (x₂,y₂), and (x₃,y₃), substituting them yields three equations:
a·x₁² + b·x₁ + c = y₁
a·x₂² + b·x₂ + c = y₂
a·x₃² + b·x₃ + c = y₃
Solving this system of three linear equations gives a, b, and c. Our calculator internally uses matrix elimination or Cramer's rule to automatically complete the calculation, simultaneously outputting the vertex coordinates (-b/(2a), (4ac-b²)/(4a)) and the axis of symmetry x = -b/(2a).
How to Use
- Enter the coordinates of the first point in the "Point 1 (x1, y1)" input fields, for example,
1 and 2. - Similarly, enter the remaining two points in "Point 2 (x2, y2)" and "Point 3 (x3, y3)", such as
2,5 and 3,10. - Click the "Solve" button, and the results area on the right will immediately display:
- Standard form:
y = 1x² + 0x + 1 - Vertex coordinates:
(0, 1) - Axis of symmetry:
x = 0 - Opening direction: Upward
- If you need to copy the results, just click the "Copy" icon next to them.
Complete Example
Problem: A parabola passes through (1,2), (2,5), and (3,10). Find its equation.
Calculation Process:
Substitute into the standard form y = ax² + bx + c:
① a·1² + b·1 + c = 2 → a + b + c = 2
② a·2² + b·2 + c = 5 → 4a + 2b + c = 5
③ a·3² + b·3 + c = 10 → 9a + 3b + c = 10
② - ① yields 3a + b = 3
③ - ① yields 8a + 2b = 8 → divide by 2: 4a + b = 4
Subtracting the two equations yields a = 1, substitute to get b = 0, then substitute into ① to get c = 1.
Result: y = x² + 1, vertex (0,1), axis of symmetry x=0, opening upward.
Interpretation: The lowest point of this parabola is (0,1), reaching a minimum value of 1 at x=0; because a>0, the graph is "U" shaped. You can enter these three sets of coordinates into the calculator to verify.
More Examples / Scenario Comparisons
Edge Case: Three Collinear Points: If you enter (0,0), (1,1), and (2,2), the calculator will show a=0, the equation degenerates into a linear function y=1x+0, and it will prompt "Three points are collinear, not a parabola." This is logical—collinear points cannot define a quadratic function.
Different Opening Direction: Entering (0,0), (1,-1), and (2,0) yields y = -x² + 2x, vertex (1,1), opening downward, indicating the highest point is (1,1).
How to Interpret the Results
| Parameter Range | Graph Characteristics | Practical Meaning |
|---|
| a > 0 | Opens upward, vertex is the minimum | e.g., Increasing returns, ascending trajectory |
| a < 0 | Opens downward, vertex is the maximum | e.g., Decreasing costs, peak prediction |
| |a| is smaller | Wider parabola | Slower rate of change |
| Δ = b²-4ac > 0 | Two x-intercepts | Equation has two real roots |
| Δ = 0 | Tangent to the x-axis | One repeated root |
| Δ < 0 | No x-intercepts | Parabola is entirely above or below the x-axis |
In our results card, besides a, b, and c, we also directly provide the vertex, axis of symmetry, and discriminant Δ to help you quickly visualize the function's graph.
Common Mistakes / Pitfalls
- Duplicate coordinates among the three points: For example, (1,2) and (1,2) are the same point, which will result in the system of equations having no unique solution.
- Confusing input order: Although the calculator automatically handles any order, it is recommended to enter them in ascending order of x to reduce confusion.
- Ignoring degeneration to a linear function when a=0: If the result shows a=0, it means the three points are collinear. In this case, the parabola "does not exist," and formulas like the axis of symmetry no longer apply.
- Confusing vertex form with standard form: This tool takes three standard points as input. If you know the vertex and another point, please use a dedicated vertex form mode (supported in future versions of this tool).
Important Notes
- This tool requires the x-coordinates of the three input points to be distinct and non-collinear; otherwise, the calculated quadratic coefficient 'a' may be inaccurate or degenerate into a linear equation.
- Coordinate values support decimals and negative numbers, but calculation precision is limited by JavaScript floating-point numbers, typically rounded to 6 decimal places.
- This tool is only applicable to the real number domain and does not handle complex roots.
- For three points with nearly equal y-coordinates (almost horizontal), minor floating-point errors may cause an incorrect sign for 'a'. It is recommended to check the precision of your raw data.
Frequently Asked Questions (FAQ)
- How many points are needed to determine a parabola?
- Generally, three non-collinear points are needed. However, for the vertex form, you only need the vertex and one additional point; for the focus-directrix form, you need the focus and the directrix.
- Why does the result say "Three points are collinear" after I enter three points?
- This indicates that all three points lie on the same straight line. In this case, the quadratic coefficient a=0, and a parabola cannot be formed. Please check if the coordinates truly reflect a quadratic relationship.
- How many decimal places are kept for a, b, and c in the results?
- We default to 4 decimal places, which is sufficient for daily use. If more precision is needed, you can manually round the raw values.
- Can this calculator directly provide the vertex form y=a(x-h)²+k?
- The current version outputs the standard form and vertex coordinates. You can write the vertex form yourself based on the vertex coordinates. Future versions will add an option to directly output the vertex form.
- How do I use the axis of symmetry to find the extreme value of a parabola?
- The axis of symmetry x = -b/(2a) is the x-coordinate of the extreme value point. Substitute it into the equation to get the y-coordinate. We already provide the vertex coordinates directly in the results, so no manual calculation is needed.
Now you can try your own parabola points in the calculator above.
