Parabola Calculator
Graph y = ax² + bx + c and find the vertex, axis of symmetry, intercepts, discriminant, vertex and factored forms, focus and directrix, step by step.
Try an example
Parabola
y = x² − 4x + 3
Opens upward (↑, a > 0) with vertex (2, −1) and two x-intercepts.
Parabola properties
- Standard form
- y = x² − 4x + 3
- Vertex (h, k)
- (2, −1)Lowest point of the curve
- Axis of symmetry
- x = 2
- Opening direction
- ↑ Upward (a > 0)
- y-intercept
- (0, 3)
- x-intercepts
- x₁ = 1, x₂ = 3
- Discriminant Δ = b² − 4ac
- 4
- Vertex form
- y = (x − 2)² − 1
- Factored form
- y = (x − 1)(x − 3)
- Minimum
- y = −1 at x = 2
- Domain
- All real numbers, (−∞, ∞)
- Range
- y ≥ −1, [−1, ∞)
The vertex is halfway between the roots: h = (x₁ + x₂)/2 = (1 + 3)/2 = 2.
What the discriminant tells you (Δ = 4)
Δ > 0 ← this equation
Two distinct real roots: the parabola crosses the x-axis twice.
Δ = 0
One repeated root: the vertex touches the x-axis.
Δ < 0
No real roots: the parabola never meets the x-axis.
Standard, vertex and factored form
| Form | Your equation | Best for |
|---|---|---|
| Standard y = ax² + bx + c | y = x² − 4x + 3 | Reading the coefficients, the y-intercept (0, c) and using the quadratic formula |
| Vertex y = a(x − h)² + k | y = (x − 2)² − 1 | Reading the vertex (h, k) and axis x = h, and seeing the graph as a shifted, stretched y = x² |
| Factored y = a(x − r₁)(x − r₂) | y = (x − 1)(x − 3) | Reading the roots (x-intercepts) directly |
Step-by-Step Solution
Step 1 — Discriminant
Δ > 0, so there are two distinct real roots.
Δ = b² − 4ac
= (−4)² − 4(1)(3)
= 16 − 12
= 4
Step 2 — Vertex
The vertex is on the axis of symmetry x = −b/(2a). Substituting it back into the equation gives the minimum value.
h = −b / (2a)
= −(−4) / (2 × 1)
= 4 / 2
= 2
k = f(2) = 2² − 4(2) + 3
= 4 − 8 + 3
= −1
Vertex = (2, −1)
Step 3 — Roots (x-intercepts)
x = (−b ± √Δ) / (2a)
= (4 ± √4) / 2
√Δ = √4 = 2
x₁ = 1
x₂ = 3
Step 4 — Vertex form
Completing the square gives y = a(x − h)² + k, using the vertex (h, k) found above.
y = a(x − h)² + k with a = 1, h = 2, k = −1
y = (x − 2)² − 1
Step 5 — Interpret the parabola
a = 1 is positive, so the parabola opens upward and the vertex is the lowest point.
Minimum: y = −1 at x = 2
Range: y ≥ −1
y-intercept: (0, 3)
Advanced parabola properties: focus and directrix
Written as (x − h)² = 4p(y − k) with p = 1/(4a) = 0.25, every point on the parabola is the same distance from the focus as from the directrix.
- Focus (h, k + p)
- (2, −0.75)
- Directrix y = k − p
- y = −1.25
How a, b and c change the graph
a
a > 0 opens upward, a < 0 downward. A larger |a| makes the parabola narrower; a smaller |a| makes it wider.
b
Together with a, b sets the axis of symmetry x = −b/(2a), so changing b moves the vertex sideways and up or down along a curve. It is also the slope of the graph where it crosses the y-axis.
c
The y-intercept: the graph crosses the y-axis at (0, c). Changing c shifts the whole parabola up or down.
Common mistakes
- Treating a = 0 as a parabola. With a = 0 there is no x² term, so y = bx + c is a straight line with no vertex.
- Getting −b/(2a) wrong. For b = −4 and a = 1, −b/(2a) = 4/2 = 2, not −2. Both the minus sign and the 2 in 2a matter.
- Forgetting the ±. The quadratic formula gives two roots, (−b + √Δ)/(2a) and (−b − √Δ)/(2a).
- Sign errors with negative b or c. Square the whole of b: (−4)² = 16. And −4ac with c = −2 becomes +8.
- Mixing up vertex and y-intercept. The y-intercept is (0, c). It is the vertex only when b = 0.
- Assuming every parabola crosses the x-axis. y = x² + 1 never does. Check the discriminant first.
- Ignoring Δ < 0. A negative discriminant means no real roots, so there are no x-intercepts to plot.
- Reading the direction backwards. a > 0 opens upward with a minimum; a < 0 opens downward with a maximum.
To solve ax² + bx + c = 0 on its own, including complex roots, use the Quadratic Formula Calculator; for cubics and higher, the Polynomial Root Calculator. The Derivative Calculator shows why the vertex is where 2ax + b = 0, and the Function Domain Calculator finds domain and range for other functions.
This parabola calculator analyses and graphs any quadratic y = ax² + bx + c. Enter the coefficients a, b and c, and it draws the parabola with its vertex, axis of symmetry, x-intercepts and y-intercept marked, choosing a viewing window that keeps all of them in sight.
It also gives the full set of parabola properties: opening direction, vertex, discriminant, roots, minimum or maximum, domain and range, plus the standard, vertex and factored forms of your equation, the focus and directrix, and a step-by-step solution. For y = x² − 4x + 3, the vertex is (2, −1), the roots are 1 and 3, and the vertex form is y = (x − 2)² − 1.
Worked Calculation Examples
| Scenario | Result | Calculation Step |
|---|---|---|
| Two real roots: y = x² − 4x + 3 | Vertex (2, −1), roots 1 and 3 | Δ = 16 − 12 = 4 > 0. h = 4/2 = 2, k = 4 − 8 + 3 = −1. x = (4 ± 2)/2 = 1 or 3. Opens upward, y-intercept (0, 3), axis x = 2, vertex form y = (x − 2)² − 1, factored form y = (x − 1)(x − 3). |
| No real roots: y = x² − 2x + 5 | Vertex (1, 4), no x-intercepts | Δ = 4 − 20 = −16 < 0, so the parabola never meets the x-axis. h = 2/2 = 1, k = 1 − 2 + 5 = 4. It opens upward from a minimum of 4, so every y-value is at least 4. Vertex form y = (x − 1)² + 4; the complex roots are 1 ± 2i. |
| Repeated root: y = x² + 4x + 4 | Vertex (−2, 0), one root x = −2 | Δ = 16 − 16 = 0, so there is one repeated root. h = −4/2 = −2 and k = 4 − 8 + 4 = 0: the vertex lies on the x-axis, where the parabola touches it. Factored form y = (x + 2)². |
| Opens downward: y = −x² + 4x − 3 | Maximum y = 1 at x = 2 | a = −1 < 0, so it opens downward. h = −4/(2 × −1) = 2, k = −4 + 8 − 3 = 1. Range y ≤ 1. Roots 1 and 3; vertex form y = −(x − 2)² + 1. |
| Decimal coefficients: y = 0.5x² − 1.5x + 1 | Vertex (1.5, −0.125), roots 1 and 2 | Δ = 2.25 − 2 = 0.25, √Δ = 0.5. h = 1.5/1 = 1.5, k = 1.125 − 2.25 + 1 = −0.125. x = (1.5 ± 0.5)/1 = 1 or 2. Factored form y = 0.5(x − 1)(x − 2). |
What Is a Parabola?
A parabola is the U-shaped (or upside-down U-shaped) curve you get when you graph a quadratic function y = ax² + bx + c with a ≠ 0. It is symmetric about a vertical line, the axis of symmetry, which passes through its turning point, the vertex.
Each coefficient has a job. a sets the direction and the width: positive a opens upward, negative a opens downward, and a larger |a| gives a narrower curve. b works together with a to place the axis of symmetry at x = −b/(2a). c is the y-intercept, the height where the graph crosses the y-axis.
Finding the Vertex, Axis and Intercepts
The axis of symmetry is x = −b/(2a), and the vertex is the point on it: h = −b/(2a) and k = f(h). For y = x² − 4x + 3, h = 4/2 = 2 and k = 4 − 8 + 3 = −1, so the vertex is (2, −1). Because a > 0, this is the minimum.
The y-intercept is always (0, c). The x-intercepts are the real roots of ax² + bx + c = 0, from the quadratic formula. When there are two, the vertex sits exactly halfway between them: h = (x₁ + x₂)/2, and (1 + 3)/2 = 2.
The Discriminant and the Number of x-Intercepts
The discriminant Δ = b² − 4ac is the part under the square root in the quadratic formula. If Δ > 0, there are two real roots and the parabola crosses the x-axis twice. If Δ = 0, there is one repeated root and the vertex just touches the x-axis. If Δ < 0, there are no real roots: the parabola stays entirely above or below the x-axis, as y = x² + 1 does.
Standard, Vertex and Factored Form
Standard form y = ax² + bx + c shows the coefficients and the y-intercept, and is what the quadratic formula uses. Vertex form y = a(x − h)² + k shows the vertex (h, k) and the axis x = h, and describes the graph as y = x² stretched by a and shifted h right and k up. Factored form y = a(x − r₁)(x − r₂) shows the roots r₁ and r₂ directly.
They are the same function written three ways: y = x² − 4x + 3 = (x − 2)² − 1 = (x − 1)(x − 3). Factored form with real factors only exists when Δ ≥ 0.
How to Use the Parabola Calculator
- Enter the coefficients a, b and c of y = ax² + bx + c. Negative numbers, decimals and fractions such as 3/4 all work; a must not be 0.
- Check the equation preview, which updates as you type (for example y = 2x² − 8x + 6).
- Read the graph: the vertex V, the dashed axis of symmetry, the x-intercepts X and the y-intercept Y are all marked. Zoom, drag to pan, or reset the view.
- Use the Parabola Properties panel for the vertex, axis, direction, intercepts, discriminant, minimum or maximum, domain and range.
- Compare the standard, vertex and factored forms of your equation, and see what the discriminant says about the roots.
- Follow the step-by-step solution: discriminant, vertex, roots, vertex form and interpretation, all calculated from your numbers.
Frequently Asked Questions
What is a parabola?
The graph of a quadratic function y = ax² + bx + c with a ≠ 0: a symmetric U-shaped curve that opens upward when a > 0 and downward when a < 0.
How do you find the vertex of a parabola?
Find h = −b/(2a), then k = f(h) by substituting h into the equation. For y = x² − 4x + 3, h = 2 and k = 2² − 4(2) + 3 = −1, so the vertex is (2, −1).
How do you find the axis of symmetry?
It is the vertical line x = −b/(2a) through the vertex. For y = x² − 4x + 3, the axis of symmetry is x = 2.
How do you find the roots of a parabola?
Solve ax² + bx + c = 0 with the quadratic formula, x = (−b ± √(b² − 4ac))/(2a), or by factoring. The real roots are the x-intercepts of the graph.
What does the discriminant tell you?
Δ = b² − 4ac gives the number of real roots: two if Δ > 0, one repeated root if Δ = 0, and none if Δ < 0. Graphically, that is how many times the parabola meets the x-axis.
How do you know whether a parabola opens upward or downward?
Look at the sign of a. If a > 0, it opens upward and the vertex is a minimum; if a < 0, it opens downward and the vertex is a maximum.
What is the vertex form of a parabola?
y = a(x − h)² + k, where (h, k) is the vertex. For example, y = x² − 4x + 3 in vertex form is y = (x − 2)² − 1. You get it by completing the square.
What is the y-intercept of a quadratic equation?
The point (0, c), found by setting x = 0 in y = ax² + bx + c. For y = x² − 4x + 3, it is (0, 3).
Can a parabola have no x-intercepts?
Yes. When the discriminant is negative, the parabola lies entirely above the x-axis (a > 0) or below it (a < 0). y = x² + 1 has no x-intercepts; its roots are the complex numbers ±i.
What is the difference between standard form and vertex form?
Standard form y = ax² + bx + c shows the coefficients and the y-intercept. Vertex form y = a(x − h)² + k shows the vertex and axis of symmetry directly. Both describe the same parabola.
How do you find the minimum or maximum of a parabola?
It is the y-coordinate k of the vertex, reached at x = h = −b/(2a). It is a minimum when a > 0 and a maximum when a < 0. For y = −x² + 4x − 3, the maximum is y = 1 at x = 2.
What are the focus and directrix of a parabola?
For y = a(x − h)² + k, let p = 1/(4a). The focus is (h, k + p) and the directrix is the line y = k − p; every point on the parabola is equally far from both. For y = x² − 4x + 3, the focus is (2, −0.75) and the directrix is y = −1.25.
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Last updated: September 27, 2026.