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Interactive parabola graph
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Formula used
Standard form
For y = ax² + bx + c, the symmetry line passes through the vertex. Dividing the negative linear coefficient by twice the quadratic coefficient gives its x-coordinate.
Vertex form
For y = a(x − h)² + k, the vertex is (h, k). The vertical line through that point is the axis of symmetry.
Factored form
When two real roots are known, their midpoint lies halfway across the parabola and therefore lies on the symmetry line.
How to use this calculator
- Select coefficients, full equation, vertex form, roots, or horizontal parabola.
- Enter all required numbers. The leading coefficient must not equal zero.
- Choose an answer format and decimal precision.
- Select “Calculate axis of symmetry.”
- Review the main answer, vertex, graph, steps, equation conversions, and verification.
- Copy, print, share, or download the completed result as needed.
Understanding the axis of symmetry
A parabola has mirror symmetry. For a vertical parabola, every point on the left side has a matching point at the same height on the right side. The vertical line separating those mirror halves is the axis of symmetry. It always passes through the vertex, which is the lowest point of an upward-opening parabola or the highest point of a downward-opening parabola.
In standard form, the coefficients do not display the vertex directly. The formula x = −b/(2a) recovers the vertex’s horizontal coordinate. After finding that value, substitute it into the original function to calculate the vertical coordinate. Together, the two values form the vertex (h, k).
Vertex form makes symmetry easy to see. In y = a(x − h)² + k, replacing x by values equally far from h produces the same squared value. Therefore, the outputs match. Factored form provides another perspective: when a parabola crosses the x-axis twice, the axis lies exactly midway between the two intercepts.
A horizontal parabola uses a horizontal symmetry line. For x = a(y − k)² + h, the axis is y = k. This calculator includes a separate mode because the roles of x and y are reversed.
Worked examples
Example 1: Standard form
For y = 2x² − 8x + 5, a = 2 and b = −8. Apply x = −b/(2a): x = −(−8)/(2 × 2) = 8/4 = 2. The axis of symmetry is x = 2. Substituting x = 2 gives y = −3, so the vertex is (2, −3).
Example 2: Vertex form
For y = −3(x + 4)² + 7, rewrite the inner expression as x − (−4). Thus h = −4 and k = 7. The axis is x = −4, the vertex is (−4, 7), and the negative coefficient shows that the parabola opens downward.
Example 3: Known roots
Suppose the roots are 1 and 9. Their midpoint is (1 + 9)/2 = 5. Therefore, the axis is x = 5. This conclusion remains valid for every nonzero leading coefficient because vertical scaling and reflection do not change the midpoint.
Example 4: Horizontal parabola
For x = 2(y − 3)² − 5, the vertex is (−5, 3). The symmetry line is horizontal and passes through the vertex, so the axis is y = 3. Since a is positive, the graph opens toward the right.
Common mistakes
Forgetting the negative sign
The standard formula is x = −b/(2a), not b/(2a). Keep the sign outside b until substitution is complete.
Dividing by only a
The denominator is 2a. Multiply a by two before dividing the numerator.
Reading vertex signs incorrectly
In (x − h), the displayed sign is opposite the value of h. For (x + 3), h equals −3.
Using a linear equation
When a equals zero, the equation is not quadratic and does not describe a parabola with this symmetry formula.
Confusing axis and vertex
The vertex is a point such as (2, −3). The axis is a complete line such as x = 2.
Assuming all roots are real
A negative discriminant gives complex roots, but the parabola still has a real vertex and a real axis of symmetry.
Frequently asked questions
What is an axis of symmetry?
It is a line that divides a graph into two mirror-image halves. A vertical parabola has an axis x = h, while a horizontal parabola has an axis y = k.
Does every quadratic function have an axis of symmetry?
Every non-degenerate real quadratic function y = ax² + bx + c with a not equal to zero has one vertical axis of symmetry.
Can a parabola have no real roots but still have an axis?
Yes. Roots describe x-axis crossings, while the axis passes through the vertex. A parabola can stay entirely above or below the x-axis and remain symmetric.
Why is the axis halfway between two roots?
The roots are points at equal height, namely y = 0. Mirror points on a parabola are equally distant from the axis, so the axis lies at their midpoint.
What happens when b equals zero?
The formula becomes x = 0. The y-axis is therefore the parabola’s axis of symmetry.
Can I use fractions or decimals?
Yes. Numeric inputs support decimal values, and the result format can display an approximate fraction, automatic decimal, fixed decimal, or scientific notation.
Is the graph exact?
The calculations use floating-point arithmetic. The graph is a visual aid and may round pixel positions, especially at extreme zoom levels.
Why must a be nonzero?
If a equals zero, the squared term disappears. The expression becomes linear or constant rather than quadratic, so it no longer produces a parabola.