Find Parabola Equation From 3 Points Calculator

Parabola Equation Calculator

Enter the coordinates of three distinct points (x1, y1), (x2, y2), and (x3, y3) to find the equation of the parabola y = ax² + bx + c passing through them.

Point	Input X	Input Y	Calculated Y (from equation)
1	1	3
2	2	8
3	3	15

What is a Find Parabola Equation from 3 Points Calculator?

A find parabola equation from 3 points calculator is a tool used to determine the equation of a quadratic function (a parabola) that passes through three given distinct points in a Cartesian coordinate system. The standard form of the parabola's equation is y = ax² + bx + c. Given three points (x1, y1), (x2, y2), and (x3, y3), the calculator solves for the coefficients a, b, and c.

This calculator is useful for students, engineers, scientists, and anyone needing to model a relationship that can be approximated by a quadratic function, based on three data points. It automates the process of solving a system of three linear equations with three variables (a, b, c).

Who should use it?

• Students: Learning algebra and coordinate geometry can use it to verify their manual calculations.
• Engineers and Scientists: For curve fitting or modeling physical phenomena (like projectile motion under certain assumptions) based on observed data points.
• Data Analysts: When trying to find a simple quadratic relationship between variables from a small set of data.

Common Misconceptions

A common misconception is that *any* three points will define a unique parabola of the form y = ax² + bx + c. This is true only if the three points have distinct x-coordinates. If two or three points share the same x-coordinate but have different y-coordinates, no such function exists. If they share the same x and y, they are the same point. Also, if the three points are collinear (lie on a straight line), the coefficient 'a' will be zero, resulting in a linear equation, not a quadratic one (a degenerate parabola).

Find Parabola Equation from 3 Points Calculator Formula and Mathematical Explanation

To find the equation of a parabola y = ax² + bx + c that passes through three points (x1, y1), (x2, y2), and (x3, y3), we substitute these points into the equation, creating a system of three linear equations:

1. y1 = a(x1)² + b(x1) + c
2. y2 = a(x2)² + b(x2) + c
3. y3 = a(x3)² + b(x3) + c

This system can be solved for a, b, and c using methods like substitution, elimination, or matrix methods (like Cramer's rule or Gaussian elimination), provided the x-coordinates are distinct.

Using Cramer's rule, we can find the determinants:

D = (x1-x2)(x1-x3)(x2-x3)

Da = y1(x2-x3) – y2(x1-x3) + y3(x1-x2)

Db = x1²(y2-y3) – x2²(y1-y3) + x3²(y1-y2)

Dc = x1²(x2*y3 – x3*y2) – x2²(x1*y3 – x3*y1) + x3²(x1*y2 – x2*y1)

If D is not zero:

a = Da / D
b = Db / D
c = Dc / D

Once a, b, and c are found, we have the equation y = ax² + bx + c. We can also find:

• Vertex: (h, k) = (-b/(2a), (4ac – b²)/(4a))
• Focus: (h, k + 1/(4a))
• Directrix: y = k – 1/(4a)

Variables Table

Variable	Meaning	Unit	Typical Range
(x1, y1), (x2, y2), (x3, y3)	Coordinates of the three points	(unitless, unitless) or units based on context	Any real numbers, but x1, x2, x3 should be distinct
a, b, c	Coefficients of the parabola equation y=ax²+bx+c	Varies	Any real numbers (a≠0 for a true parabola)
h, k	Coordinates of the vertex	Same as x, y	Any real numbers

Practical Examples (Real-World Use Cases)

Example 1: Projectile Path

Imagine a ball is thrown, and its height is measured at three different horizontal distances: at 1 meter it's 3 meters high, at 2 meters it's 8 meters high, and at 3 meters it's 11 meters high (a very strange throw, maybe it was boosted). Let's find the parabolic path y = ax² + bx + c.

• Point 1: (1, 3)
• Point 2: (2, 8)
• Point 3: (3, 11)

Using the find parabola equation from 3 points calculator with these inputs (1,3), (2,8), (3,11), we might get a = -1, b = 8, c = -4. So, the equation would be y = -x² + 8x – 4. The vertex would be at x = -8/(2*-1) = 4, and y = -(4)² + 8(4) – 4 = -16 + 32 – 4 = 12. Vertex (4, 12).

Example 2: Cost Function

A company finds that producing 10 units costs $300, 20 units costs $400, and 30 units costs $700. They suspect the cost function C(x) might be quadratic, C(x) = ax² + bx + c, where x is the number of units.

• Point 1: (10, 300)
• Point 2: (20, 400)
• Point 3: (30, 700)

Plugging these into the find parabola equation from 3 points calculator, we'd get values for a, b, and c, giving us the quadratic cost function. For instance, if a=1, b=-10, c=300, the equation is C(x) = x² – 10x + 300.

How to Use This Find Parabola Equation from 3 Points Calculator

1. Enter Point 1: Input the x-coordinate (x1) and y-coordinate (y1) of the first point.
2. Enter Point 2: Input the x-coordinate (x2) and y-coordinate (y2) of the second point.
3. Enter Point 3: Input the x-coordinate (x3) and y-coordinate (y3) of the third point. Ensure the x-coordinates are distinct for a unique parabola of the form y=ax²+bx+c.
4. Calculate: Click the "Calculate" button or simply change the input values. The results will update automatically.
5. Read Results: The calculator will display the equation y = ax² + bx + c with the calculated values of a, b, and c. It will also show the vertex, focus, and directrix.
6. Check Table and Chart: The table verifies the equation by plugging the input x-values back into it, and the chart visualizes the parabola and the points.
7. Reset: Click "Reset" to clear the inputs to their default values.
8. Copy: Click "Copy Results" to copy the main equation and key values to your clipboard.

The find parabola equation from 3 points calculator provides immediate feedback. If the points are collinear or have non-distinct x-values (for this form), it will indicate an issue.

Key Factors That Affect Find Parabola Equation from 3 Points Calculator Results

1. Coordinates of the Points (x1, y1), (x2, y2), (x3, y3): These are the primary inputs. The specific values directly determine the coefficients a, b, and c.
2. Distinctness of x-coordinates: If x1=x2, x1=x3, or x2=x3, you cannot form a unique parabola of the form y=ax²+bx+c unless the corresponding y-values are also identical (and even then, it doesn't define a unique parabola without more constraints, or no parabola if y's differ). The calculator assumes distinct x-values for a non-vertical parabola.
3. Collinearity of the Points: If the three points lie on a straight line, the coefficient 'a' will be zero, and the equation becomes linear (y=bx+c), not quadratic. The calculator might indicate this or return a very small 'a'.
4. Magnitude of Coordinates: Very large or very small coordinate values can lead to very large or small coefficients, potentially affecting numerical precision in some calculations.
5. Orientation of the Parabola: This calculator assumes the parabola opens upwards or downwards (y=ax²+bx+c). If the points define a parabola opening sideways (x=ay²+by+c), this form won't fit, and you'd need a different approach or to swap x and y if appropriate.
6. Precision of Input: Small changes or errors in the input coordinates can lead to different resulting equations, especially if the points are close together or nearly collinear.

Frequently Asked Questions (FAQ)

1. What if the three points lie on a straight line?

If the points are collinear, the coefficient 'a' in y=ax²+bx+c will be zero (or very close to zero due to rounding), and the equation will be linear (y=bx+c). The find parabola equation from 3 points calculator might indicate this or simply show a=0.

2. What if two of the x-coordinates are the same?

If, for example, x1=x2 but y1≠y2, then no function y=ax²+bx+c can pass through these points because a function cannot have two different y-values for the same x-value. The calculator will likely give an error or undefined result as the denominator in the formulas for a, b, and c would be zero.

3. Can I find the equation of a parabola that opens sideways?

This specific calculator finds the equation y=ax²+bx+c, which opens up or down. To find one opening sideways (x=ay²+by+c), you would swap the roles of x and y in your input points and solve for x in terms of y.

4. How accurate is the find parabola equation from 3 points calculator?

It's as accurate as the underlying arithmetic and the precision of the input values. For standard numerical inputs, it should be very accurate.

5. What does the vertex of the parabola represent?

The vertex is the point where the parabola changes direction – either the minimum point (if it opens upwards, a>0) or the maximum point (if it opens downwards, a<0).

6. What are the focus and directrix?

The focus is a point, and the directrix is a line. A parabola is defined as the set of all points that are equidistant from the focus and the directrix.

7. Can I use this calculator for more than three points?

No, this calculator is specifically for finding the unique quadratic passing through exactly three non-collinear points with distinct x-values. For more points, you'd look into methods like least squares regression to find the best-fit parabola. See our curve fitting tools.

8. What if the calculator gives 'Infinity' or 'NaN'?

This usually means the denominator 'D' in the formulas was zero, which happens if the x-coordinates are not distinct or if there's an issue with collinearity preventing a unique quadratic solution of the form y=ax²+bx+c.

Related Tools and Internal Resources

• Quadratic Equation Solver: Solve equations of the form ax² + bx + c = 0.
• Vertex Calculator: Find the vertex of a parabola given its equation.
• Focus and Directrix Calculator: Calculate the focus and directrix from the parabola's equation.
• Graphing Calculator: Plot various functions, including parabolas.
• System of Linear Equations Solver: Solve systems of linear equations like the one used here.
• Curve Fitting Tools: Find curves that best fit a set of data points, including quadratic fits.