Calculated results
f(x) = x² - 4x + 3Interactive graph
Transformations from y = x²
- Shifted 2 units right
- Shifted 1 units down
Step-by-step solution
- Read the coefficients: a = 1, b = -4, and c = 3.
- Calculate the discriminant: Δ = b² − 4ac = (-4)² − 4(1)(3) = 4.
- Apply the quadratic formula: x = [−b ± √Δ] / 2a.
- The roots are x₁ = 3 and x₂ = 1.
- Find the vertex x-coordinate: h = −b / 2a = 2.
- Evaluate f(h): k = -1. The vertex is (2, -1).
- The axis of symmetry is x = 2.
- The y-intercept is f(0) = c = 3.
- The focus is (h, k + 1/4a), and the directrix is y = k − 1/4a.
Coordinate table
| Point | x | f(x) | Symmetry note |
|---|---|---|---|
| 1 | -2 | 15 | Distance from axis: 4 |
| 2 | -1 | 8 | Distance from axis: 3 |
| 3 | 0 | 3 | Distance from axis: 2 |
| 4 | 1 | 0 | Distance from axis: 1 |
| 5 | 2 | -1 | Vertex |
| 6 | 3 | 0 | Distance from axis: 1 |
| 7 | 4 | 3 | Distance from axis: 2 |
| 8 | 5 | 8 | Distance from axis: 3 |
| 9 | 6 | 15 | Distance from axis: 4 |
Quadratic function inputs
Formula used
How to use this calculator
- Select coefficient entry or complete equation entry.
- Enter the quadratic function using supported notation.
- Choose graph ranges, precision, rows, and step size.
- Enable comparison mode when studying two parabolas.
- Press Calculate and graph to view every result.
- Use the export buttons to save your work.
Example data
| Equation | Vertex | Roots | Opening | Range |
|---|---|---|---|---|
| x² − 4x + 3 | (2, −1) | 1, 3 | Upward | y ≥ −1 |
| 2(x − 3)² − 8 | (3, −8) | 1, 5 | Upward | y ≥ −8 |
| −(x − 2)(x + 4) | (−1, 9) | −4, 2 | Downward | y ≤ 9 |
Frequently asked questions
What is a quadratic function?
A quadratic function has highest exponent two. Its graph forms a parabola. The leading coefficient cannot equal zero.
What does the coefficient a control?
Coefficient a controls direction and width. Positive values open the graph upward. Negative values open it downward.
How is the vertex calculated?
Calculate h using negative b over two a. Then evaluate the function at h. The resulting point is the vertex.
What does the discriminant show?
The discriminant classifies the quadratic roots. Positive values produce two real roots. Negative values produce complex conjugates.
What is the axis of symmetry?
It is the vertical line through the vertex. The parabola mirrors across this line. Its equation is x equals h.
Can the calculator graph vertex form?
Yes, it recognizes common vertex notation. Parentheses and squared notation must be entered correctly. The calculator converts it automatically.
Can it compare two quadratic functions?
Yes, enable the comparison switch. Both functions appear on one graph. Real intersection points are calculated automatically.
Why are roots sometimes complex?
Complex roots occur when the discriminant is negative. The parabola misses the x-axis entirely. Conjugate imaginary solutions still exist.
How do I download the graph?
Use the Download PNG button below the graph. The current graph view becomes an image. Your browser saves the file.