Desmos Parabola Equation Calculator
This calculator helps you find the equation of a parabola in vertex form (y = a(x-h)² + k) given its vertex and one other point it passes through. You can then easily input this equation into Desmos to visualize the graph.
' + equation + '' +
'You can copy this equation directly into Desmos to graph your parabola.';
}
Understanding Parabolas and Desmos
Desmos is a powerful and intuitive online graphing calculator that allows users to visualize mathematical functions, explore data, and create interactive graphs. While Desmos excels at plotting equations, sometimes the challenge lies in finding the correct equation for a specific graph you have in mind or from given points. This calculator aims to bridge that gap for parabolas.
The Vertex Form of a Parabola: y = a(x-h)² + k
A parabola is a U-shaped curve that is the graph of a quadratic function. One of the most useful forms for understanding and graphing parabolas is the vertex form:
y = a(x-h)² + k
- (h, k) represents the coordinates of the vertex of the parabola. The vertex is the turning point of the parabola – either its lowest point (minimum) or highest point (maximum).
- a is a coefficient that determines the parabola's direction and vertical stretch or compression.
- If
a > 0, the parabola opens upwards. - If
a < 0, the parabola opens downwards. - The larger the absolute value of
a, the narrower the parabola. - The smaller the absolute value of
a(closer to zero), the wider the parabola.
- If
Knowing the vertex and just one other point on the parabola is enough to uniquely determine its equation in this form.
How This Calculator Works
This calculator takes four inputs:
- The X-coordinate of the parabola's vertex (
h). - The Y-coordinate of the parabola's vertex (
k). - The X-coordinate of any other point that lies on the parabola (
x₁). - The Y-coordinate of that same other point (
y₁).
Using these inputs, it calculates the value of the 'a' coefficient. The formula used is derived by substituting the vertex (h, k) and the other point (x₁, y₁) into the vertex form equation:
y₁ = a(x₁ - h)² + k
Solving for a gives:
a = (y₁ - k) / (x₁ - h)²
Once 'a' is found, the calculator constructs the complete vertex form equation, which you can then directly paste into Desmos.
Using the Calculator for Desmos Graphing
- Identify your points: Determine the vertex of your desired parabola and at least one other point it should pass through.
- Input coordinates: Enter these four coordinate values into the respective fields in the calculator.
- Calculate: Click the "Calculate Equation" button.
- Get the equation: The calculator will display the calculated 'a' value and the full parabola equation in vertex form.
- Graph in Desmos: Copy the generated equation (e.g.,
y = 2(x - 3)² + 1) and paste it directly into the Desmos input bar. Desmos will instantly graph your parabola!
Examples
Example 1: Basic Parabola
- Vertex: (0, 0)
- Another Point: (1, 1)
- Calculation:
a = (1 - 0) / (1 - 0)² = 1 / 1 = 1 - Resulting Equation:
y = x² - In Desmos: Type
y = x^2
Example 2: Shifted and Stretched Parabola
- Vertex: (2, 3)
- Another Point: (3, 5)
- Calculation:
a = (5 - 3) / (3 - 2)² = 2 / 1² = 2 - Resulting Equation:
y = 2(x - 2)² + 3 - In Desmos: Type
y = 2(x - 2)^2 + 3
Example 3: Downward-Opening Parabola
- Vertex: (-1, 4)
- Another Point: (0, 3)
- Calculation:
a = (3 - 4) / (0 - (-1))² = -1 / (1)² = -1 - Resulting Equation:
y = -(x + 1)² + 4 - In Desmos: Type
y = -(x + 1)^2 + 4
This calculator simplifies the process of finding specific parabola equations, making your Desmos graphing experience more efficient and precise.