Find Equation Of Parabola Calculator

Enter the coordinates of the vertex (h, k) and one other point (x, y) on the parabola to find its equation in vertex and standard form using our find equation of parabola calculator.

Parabola Equation Calculator

X	Y (Calculated)
Enter values to see table data.

Table of points on the calculated parabola.

What is a Find Equation of Parabola Calculator?

A find equation of parabola calculator is a tool designed to determine the mathematical equation that describes a parabola, given specific information about it. Most commonly, if you know the coordinates of the parabola's vertex (the turning point) and the coordinates of at least one other point that lies on the parabola, this calculator can derive the equation in both vertex form (y = a(x-h)² + k) and standard form (y = ax² + bx + c). This is particularly useful in algebra, physics, and engineering where parabolas model various phenomena.

Anyone studying quadratic functions, conic sections, or dealing with projectile motion, satellite dish design, or optical systems might use a find equation of parabola calculator. It saves time and reduces the chance of algebraic errors when deriving the equation manually.

A common misconception is that any three points will define a unique parabola opening up or down. While three non-collinear points define a unique parabola, our calculator specifically uses the vertex and one other point, which is a common and efficient method.

Find Equation of Parabola Formula and Mathematical Explanation

When the vertex (h, k) and another point (x, y) on the parabola are known, we start with the vertex form of the parabola's equation:

y = a(x – h)² + k

Where:

• (h, k) are the coordinates of the vertex.
• (x, y) are the coordinates of the other point on the parabola.
• 'a' is a coefficient that determines the parabola's width and direction (up or down).

Step-by-step derivation:

1. Substitute the known values of h, k, x, and y into the vertex form equation: y_point = a(x_point – h)² + k.
2. Solve for 'a': y_point – k = a(x_point – h)² a = (y_point – k) / (x_point – h)² (provided x_point ≠ h)
3. Once 'a' is found, you have the equation in vertex form: y = a(x – h)² + k.
4. To get the standard form (y = ax² + bx + c), expand the vertex form: y = a(x² – 2hx + h²) + k y = ax² – 2ahx + ah² + k So, b = -2ah and c = ah² + k.

Variable	Meaning	Unit	Typical Range
h	x-coordinate of the vertex	(length units)	Any real number
k	y-coordinate of the vertex	(length units)	Any real number
x	x-coordinate of the given point	(length units)	Any real number
y	y-coordinate of the given point	(length units)	Any real number
a	Leading coefficient (stretch/compression and direction)	(units of y / units of x²)	Any non-zero real number
b	Coefficient of x in standard form	(units of y / units of x)	Any real number
c	y-intercept in standard form	(units of y)	Any real number

Practical Examples (Real-World Use Cases)

Let's see how the find equation of parabola calculator works with practical examples.

Example 1: Projectile Motion

Imagine a ball is thrown, and its path is a parabola. Suppose the highest point (vertex) it reaches is (3, 10) meters, and it passes through the point (5, 6) meters.

• Vertex (h, k) = (3, 10)
• Point (x, y) = (5, 6)

Using the calculator or formulas: a = (6 – 10) / (5 – 3)² = -4 / 4 = -1.

Vertex form: y = -1(x – 3)² + 10

Standard form: y = -(x² – 6x + 9) + 10 = -x² + 6x – 9 + 10 = -x² + 6x + 1

The equation of the ball's path is y = -x² + 6x + 1.

Example 2: Satellite Dish Cross-Section

A satellite dish has a parabolic cross-section. Let's say the vertex is at (0, 0), and the dish passes through the point (2, 1) (where units are feet).

• Vertex (h, k) = (0, 0)
• Point (x, y) = (2, 1)

a = (1 – 0) / (2 – 0)² = 1 / 4 = 0.25

Vertex form: y = 0.25(x – 0)² + 0 = 0.25x²

Standard form: y = 0.25x²

The equation for the dish's cross-section is y = 0.25x².

How to Use This Find Equation of Parabola Calculator

1. Enter Vertex Coordinates: Input the x-coordinate (h) and y-coordinate (k) of the parabola's vertex into the respective fields.
2. Enter Point Coordinates: Input the x-coordinate (x) and y-coordinate (y) of the other point that lies on the parabola.
3. View Results: The calculator will automatically update and display the value of 'a', the equation in vertex form, the values of 'b' and 'c', and the equation in standard form. The graph and table will also update.
4. Check for Errors: If you enter non-numeric values or if the x-coordinate of the point is the same as the vertex, an error message will guide you.
5. Interpret the Graph and Table: The graph shows the shape of your parabola, the vertex, and the given point. The table lists some points on the parabola around the vertex.
6. Reset or Copy: Use the "Reset" button to clear inputs to default values or "Copy Results" to copy the main equations and values.

The primary result is the standard form equation, which is widely used. The vertex form is useful for quickly identifying the vertex.

Key Factors That Affect Parabola Equation Results

Several factors influence the equation derived by the find equation of parabola calculator:

• Vertex Position (h, k): The location of the vertex directly sets the 'h' and 'k' in the vertex form and influences 'b' and 'c' in the standard form.
• Position of the Other Point (x, y): The coordinates of this point, relative to the vertex, determine the value of 'a'.
• Value of 'a': This coefficient dictates how wide or narrow the parabola is and whether it opens upwards (a > 0) or downwards (a < 0). It's calculated based on the vertex and the point.
• Horizontal Distance (x-h): The horizontal distance between the point and the vertex. If this is zero, 'a' is undefined in our initial formula (it would be a vertical line, not a function y=ax²+bx+c).
• Vertical Distance (y-k): The vertical distance between the point and the vertex, relative to the horizontal distance squared, defines 'a'.
• Accuracy of Input Values: Small changes in the input coordinates can lead to different equations, especially affecting 'a', 'b', and 'c'.

Frequently Asked Questions (FAQ)

What is a parabola?

A parabola is a U-shaped curve that is a graph of a quadratic equation (y = ax² + bx + c). It is also defined as the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).

What is the vertex of a parabola?

The vertex is the point where the parabola changes direction; it's the minimum point if the parabola opens upwards or the maximum point if it opens downwards.

What if the x-coordinate of the point is the same as the vertex?

If the x-coordinate of the point (x) is the same as the x-coordinate of the vertex (h), and the y-coordinates are different, then 'a' cannot be found using the y=a(x-h)²+k form directly because (x-h)² would be zero, leading to division by zero. This scenario implies a vertical line if the y-values differ, which isn't a standard parabola function y=ax²+bx+c. Our find equation of parabola calculator will show an error.

Can I find the equation with three random points?

Yes, but that requires a different method (solving a system of three linear equations for a, b, and c). This calculator is specifically for when you know the vertex and one other point.

How do I know if the parabola opens up or down?

The sign of 'a' tells you. If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards. The find equation of parabola calculator calculates 'a'.

What is the axis of symmetry?

The axis of symmetry is a vertical line that passes through the vertex, x = h. The parabola is symmetrical about this line.

How does the value of 'a' affect the shape?

If |a| > 1, the parabola is narrower (vertically stretched) than y=x². If 0 < |a| < 1, it is wider (vertically compressed). The find equation of parabola calculator finds 'a' for you.

Can I use this calculator for horizontal parabolas?

No, this calculator is for parabolas that open up or down, represented by y = ax² + bx + c. Horizontal parabolas have the form x = ay² + by + c.