Calculator
Graph
Example Data Table
| a | b | c | Vertex | Direction |
|---|---|---|---|---|
| 1 | -4 | 3 | (2, -1) | Upward |
| -2 | 8 | 1 | (2, 9) | Downward |
| 0.5 | -3 | 2 | (3, -2.5) | Upward |
Formula Used
For a quadratic equation:
y = ax² + bx + c
The vertex x-coordinate is:
h = -b / 2a
The vertex y-coordinate is:
k = ah² + bh + c
The axis of symmetry is x = h. If a is positive, the parabola opens upward. If a is negative, it opens downward.
How to Use This Calculator
Enter the values of a, b, and c from your quadratic equation. Make sure the equation follows standard form. The calculator finds the vertex, axis of symmetry, opening direction, discriminant, roots, and selected function value. Press the calculate button to see results above the form and view the graph.
Understanding the Vertex of a Parabola
The vertex is the turning point of a parabola. It is the lowest point when the parabola opens upward. It is the highest point when the parabola opens downward.
Why the Vertex Matters
The vertex helps explain the main behavior of a quadratic equation. It shows where the curve changes direction. This point is useful in algebra, physics, design, business, and optimization problems.
Standard Quadratic Form
A parabola is often written as y = ax² + bx + c. The value of a controls the opening direction and curve width. The value of b shifts the curve sideways. The value of c gives the y-intercept.
Minimum and Maximum Values
When a is positive, the vertex gives the minimum value. When a is negative, the vertex gives the maximum value. This makes the calculator useful for profit, height, area, and motion questions.
Axis of Symmetry
The axis of symmetry is a vertical line through the vertex. It divides the parabola into two matching halves. Its equation is x = -b / 2a.
Graph Support
The graph helps users see the curve visually. It plots points around the vertex. This makes the result easier to understand and verify.
Advanced Result Details
This tool also calculates the discriminant and real roots. The discriminant shows how many real x-intercepts exist. A positive value means two real roots. A zero value means one repeated root. A negative value means no real roots.
FAQs
What is the vertex of a parabola?
The vertex is the turning point of the parabola. It is either the lowest or highest point, depending on the sign of coefficient a.
What formula finds the vertex?
Use h = -b / 2a. Then place h into y = ax² + bx + c to find k.
Can a be zero?
No. If a is zero, the equation is linear. A parabola requires a nonzero quadratic coefficient.
What does the axis of symmetry mean?
It is the vertical line passing through the vertex. It divides the parabola into two equal mirror sides.
How do I know if it opens upward?
If a is positive, the parabola opens upward. The vertex then gives the minimum value.
How do I know if it opens downward?
If a is negative, the parabola opens downward. The vertex then gives the maximum value.
What does the discriminant show?
The discriminant shows the number of real roots. Positive means two, zero means one, and negative means none.
Can I download my result?
Yes. Use the CSV or PDF download buttons after calculating your parabola result.