Axis of Symmetry Calculator
Instantly find the axis of symmetry for a parabola. Our calculator handles standard, vertex, and intercept forms. Shows the formula and step-by-step work.
Equation Inputs
Switch between the equation forms and enter the coefficients. The active form controls how the axis of symmetry is calculated.
Standard form: y = ax² + bx + c. Provide all three coefficients and the calculator applies x = -b / (2a).
Vertex form: y = a(x - h)² + k. The axis of symmetry is the vertical line x = h.
Intercept form: y = a(x - p)(x - q). Use the midpoint formula x = (p + q) / 2.
Step-by-step
How to Use This Calculator
This tool supports the three common quadratic representations. Switch the tab to focus on your equation form, fill in the coefficients, and click Calculate to see the axis of symmetry and a transparent, numbered walkthrough.
Methodology
The calculator uses the standard axis formulas for each form and interprets extra formatting (fractions or decimals) safely. Results are deterministic: the same inputs always produce the same rounded axis, and the steps mirror the math shown to you.
1. Standard Form: y = ax² + bx + c
The axis of symmetry is the vertical line that passes through the vertex. Compute it with x = -b / (2a). The value of a must not be zero or the equation is not a parabola.
Example: For y = 2x² + 8x - 5, plug in a=2, b=8:
Result: x = -8 / (2×2) = -8 / 4 = -2. The axis is x = -2.
2. Vertex Form: y = a(x - h)² + k
The vertex explicit in this form gives the axis directly: x = h. The calculator reads your h value and displays this line.
Example: For y = 3(x - 4)² + 1, the axis of symmetry is x = 4 (if the equation were y = 3(x + 4)² + 1, interpret h = -4).
3. Intercept Form: y = a(x - p)(x - q)
The axis runs halfway between the x-intercepts p and q. Use the midpoint formula x = (p + q) / 2 to find it.
Example: For y = -2(x - 1)(x - 7), the intercepts are p=1 and q=7, so x = (1 + 7) / 2 = 4.
Frequently Asked Questions (FAQ)
What is the axis of symmetry?
The axis of symmetry is a vertical line that divides a parabola into two perfect mirror images. It passes through the vertex, the highest or lowest point of the curve.
How is the axis of symmetry related to the vertex of a parabola?
The axis of symmetry is the x-coordinate of the vertex. If the vertex is at (h, k), the axis is x = h, and this calculator identifies that value.
What is the axis of symmetry for a horizontal parabola?
Horizontal parabolas (functions of y) are not supported here. Their axis is horizontal: y = -b / (2a). This calculator focuses on vertical parabolas (functions of x).
What does the 'a' value do?
The 'a' coefficient determines the opening direction and steepness. If a is positive, the parabola opens upward; if negative, it opens downward. In vertex and intercept forms, a does not shift the axis, but it is essential for standard form calculations.
Full original guide (expanded)
About the author
Ugo Candido builds algebra & financial tools to help readers make better decisions with transparent models. Each calculator reflects the mathematical steps a teacher or lender would show when calculating solutions.
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