Find Quadratic Equation from Vertex and Point Calculator
Calculator
Enter the vertex (h, k) and another point (x, y) on the parabola to find its equation.
Results:
Graph of the calculated parabola.
| Parameter | Value |
|---|---|
| Vertex h | 2 |
| Vertex k | 3 |
| Point x | 4 |
| Point y | 7 |
| a | |
| b | |
| c |
Table of input values and calculated coefficients.
What is a Find Quadratic Equation from Vertex and Point Calculator?
A find quadratic equation from vertex and point calculator is a tool used to determine the equation of a parabola (a quadratic function) when you know the coordinates of its vertex (h, k) and one other point (x, y) that lies on the parabola. The calculator uses these two pieces of information to find the specific quadratic equation in both vertex form, y = a(x – h)² + k, and standard form, y = ax² + bx + c. Our find quadratic equation from vertex and point calculator automates this process.
This calculator is particularly useful for students learning algebra, teachers creating examples, engineers, and anyone needing to model a parabolic curve based on these specific known points. By inputting the vertex and a point, the find quadratic equation from vertex and point calculator quickly provides the value of 'a' and the complete equations.
Common misconceptions include thinking that any two points are enough (you specifically need the vertex and one other point, or three non-collinear points) or that the 'a' value is always 1 (it is determined by the vertex and the other point).
Find Quadratic Equation from Vertex and Point Formula and Mathematical Explanation
The vertex form of a quadratic equation is given by:
y = a(x – h)² + k
where (h, k) is the vertex of the parabola, and 'a' is a constant that determines the parabola's direction and width.
If we are given the vertex (h, k) and another point (x, y) on the parabola, we can substitute these values into the vertex form to solve for 'a':
- Start with the vertex form: y = a(x – h)² + k
- Substitute the coordinates of the given point (x, y) and the vertex (h, k) into the equation: ypoint = a(xpoint – h)² + k
- Rearrange to solve for 'a': ypoint – k = a(xpoint – h)² a = (ypoint – k) / (xpoint – h)² (provided xpoint ≠ h)
Once 'a' is found, you have the equation in vertex form. To get the standard form y = ax² + bx + c, you expand the vertex form:
y = a(x² – 2hx + h²) + k
y = ax² – 2ahx + ah² + k
So, b = -2ah and c = ah² + k. Our find quadratic equation from vertex and point calculator performs these steps automatically.
Variables Table
| Variable | Meaning | Unit | Typical Range |
|---|---|---|---|
| h | x-coordinate of the vertex | None (coordinate) | Any real number |
| k | y-coordinate of the vertex | None (coordinate) | Any real number |
| x | x-coordinate of the given point | None (coordinate) | Any real number (≠ h) |
| y | y-coordinate of the given point | None (coordinate) | Any real number |
| a | Coefficient determining width and direction | None | Any non-zero real number |
| b | Coefficient of x in standard form | None | Any real number |
| c | Constant term (y-intercept) in standard form | None | Any real number |
Practical Examples (Real-World Use Cases)
Let's see how the find quadratic equation from vertex and point calculator works with examples.
Example 1: Projectile Motion
Imagine a ball is thrown, and its path is a parabola. The highest point (vertex) it reaches is at (3, 10) – 3 seconds after launch, 10 meters high. At 5 seconds (x=5), the ball is at a height of 6 meters (y=6). We want to find the equation of its path.
- Vertex (h, k) = (3, 10)
- Point (x, y) = (5, 6)
Using the formula a = (y – k) / (x – h)² = (6 – 10) / (5 – 3)² = -4 / 2² = -4 / 4 = -1.
So, a = -1. The vertex form is y = -1(x – 3)² + 10.
The standard form is y = -(x² – 6x + 9) + 10 = -x² + 6x – 9 + 10 = -x² + 6x + 1.
Example 2: Parabolic Arch
A parabolic arch has its vertex at (0, 20) (meaning its highest point is 20 units above the origin). The arch touches the ground at (10, 0).
- Vertex (h, k) = (0, 20)
- Point (x, y) = (10, 0)
a = (0 – 20) / (10 – 0)² = -20 / 100 = -0.2.
Vertex form: y = -0.2(x – 0)² + 20 = -0.2x² + 20.
Standard form: y = -0.2x² + 0x + 20.
You can verify these with our find quadratic equation from vertex and point calculator.
How to Use This Find Quadratic Equation from Vertex and Point Calculator
Using our find quadratic equation from vertex and point calculator is straightforward:
- Enter Vertex Coordinates: Input the h-value (x-coordinate) and k-value (y-coordinate) of the parabola's vertex into the "Vertex (h)" and "Vertex (k)" fields respectively.
- Enter Point Coordinates: Input the x-value and y-value of another point on the parabola into the "Point (x)" and "Point (y)" fields. Make sure the x-value of the point is different from the x-value of the vertex.
- View Results: The calculator will instantly display the calculated value of 'a', 'b', and 'c', the equation in vertex form (y = a(x-h)² + k), and the equation in standard form (y = ax² + bx + c). The results table and the graph will also update.
- Interpret the Graph: The graph visually represents the parabola based on your inputs, with the vertex and the given point highlighted.
- Reset or Copy: Use the "Reset" button to clear the fields to their default values or the "Copy Results" button to copy the inputs and calculated equations.
The find quadratic equation from vertex and point calculator helps you visualize and understand the relationship between the vertex, a point, and the resulting parabolic equation.
Key Factors That Affect Find Quadratic Equation from Vertex and Point Results
The resulting quadratic equation is directly influenced by the input values:
- Vertex Position (h, k): The vertex determines the axis of symmetry (x=h) and the minimum or maximum value (k) of the quadratic function. Changing h shifts the parabola horizontally, and changing k shifts it vertically.
- Point Position (x, y): The coordinates of the other point, relative to the vertex, determine the 'a' value.
- Difference (x – h): The horizontal distance between the point and the vertex. The square of this difference is in the denominator for 'a'. If (x-h) is small, and (y-k) is not, 'a' will be large (narrow parabola). If x=h, the 'a' value is undefined, as a vertical line cannot be represented by y=f(x) in this way, and the point would be vertically aligned with the vertex, but different, which is impossible for a function.
- Difference (y – k): The vertical distance between the point and the vertex. This is in the numerator for 'a'. If y=k and x≠h, then a=0, which means it's a horizontal line, not a quadratic (our calculator will show a=0).
- Sign of (y – k): If y > k and the point is above the vertex, and the parabola opens upwards or downwards depending on context (but 'a' will be positive if (x-h)² is positive). If y < k, 'a' will be negative (assuming (x-h)² is positive), and the parabola opens downwards.
- Magnitude of 'a': The value of 'a' calculated from a = (y – k) / (x – h)² determines the "steepness" or "width" of the parabola. A larger |a| means a narrower parabola, and a smaller |a| means a wider parabola. The sign of 'a' determines if it opens upwards (a > 0) or downwards (a < 0). Using a quadratic equation calculator after finding the equation can help find roots.
Our find quadratic equation from vertex and point calculator accurately reflects these factors.
Frequently Asked Questions (FAQ)
What if the given point is the vertex itself?
If you input the vertex coordinates as the point coordinates (x=h, y=k), then (x-h)=0 and (y-k)=0. The formula for 'a' becomes 0/0, which is indeterminate. You need a point *other* than the vertex to uniquely define the parabola's 'a' value. The calculator will likely show an error or NaN if x=h.
Can I find the equation if I have the vertex and the y-intercept?
Yes, the y-intercept is a point where x=0. So, if the y-intercept is (0, c), you have a point (0, c) and the vertex (h, k). You can use these in the find quadratic equation from vertex and point calculator.
What does it mean if 'a' is zero?
If 'a' is calculated as zero, it means y – k = 0, so y = k, provided x ≠ h. This would imply the "parabola" is actually a horizontal line y = k, which is not a quadratic equation. This happens if the 'point y' is the same as 'vertex k'.
How is the standard form y = ax² + bx + c derived from the vertex form?
By expanding y = a(x – h)² + k: y = a(x² – 2hx + h²) + k = ax² – 2ahx + ah² + k. So, b = -2ah and c = ah² + k. See our standard form of quadratic equation converter for more.
Can I use this find quadratic equation from vertex and point calculator for parabolas opening left or right?
No, this calculator finds equations of the form y = f(x), which are parabolas opening upwards or downwards. Parabolas opening left or right have the form x = a(y – k)² + h.
What if the x-coordinate of the point is the same as the x-coordinate of the vertex (x=h)?
If x=h and y≠k, you have two points vertically aligned, which cannot happen in a function y=f(x). The denominator (x-h)² becomes zero, and 'a' is undefined. Our find quadratic equation from vertex and point calculator will indicate an error or NaN for 'a' if x=h.
Does the order of vertex and point matter in the find quadratic equation from vertex and point calculator?
You must correctly identify which is the vertex (h, k) and which is the other point (x, y) for the formula a = (y – k) / (x – h)² to be applied correctly.
Where is the axis of symmetry?
The axis of symmetry is a vertical line passing through the vertex, given by the equation x = h. Explore more with a parabola equation calculator.
Related Tools and Internal Resources
- Quadratic Equation Solver: Solves for the roots of a standard quadratic equation ax² + bx + c = 0.
- Vertex Calculator: Finds the vertex (h, k) of a quadratic equation given in standard form.
- Parabola Grapher: Graphs quadratic equations and shows key features like vertex and intercepts.
- Standard Form Converter: Converts quadratic equations between vertex and standard forms.
- Algebra Calculators: A collection of calculators for various algebra problems.
- Math Tools: More general math and geometry tools.