Calculator Inputs
Formula Used
y = a(x - h)² + k
h = -b/(2a), k = f(h)
Δ = b² - 4ac
p = 1/(4a), focus = (h, k + p), directrix: y = k - p
How to Use This Calculator
- Select the form or data you already know.
- Enter coefficients, roots, a vertex and point, or three coordinates.
- Choose decimal or fractional formatting and set precision.
- Adjust graph limits when the parabola needs a wider viewing window.
- Click the calculation button to see equations, roots, geometry, transformations, and steps.
- Copy, print, export CSV, or save the graph image.
Example Data
| Mode | Input | Vertex | Vertex form |
|---|---|---|---|
| Standard form | a = 1, b = -4, c = 3 | (2, -1) | y = (x - 2)² - 1 |
| Vertex form | a = -2, h = 1, k = 5 | (1, 5) | y = -2(x - 1)² + 5 |
| Factored form | a = 1, r₁ = -1, r₂ = 5 | (2, -9) | y = (x - 2)² - 9 |
| Vertex and point | Vertex (0, 1), point (2, 9) | (0, 1) | y = 2x² + 1 |
| Three points | (0, 3), (1, 0), (3, 0) | (2, -1) | y = (x - 2)² - 1 |
Understanding Vertex Form
Vertex form reveals a parabola’s most important geometric information immediately. The point (h, k) is the turning point, while a controls direction and width. A positive value opens upward, and a negative value opens downward.
The absolute value of a compares the graph with y = x². Values above one create a narrower parabola, while values between zero and one create a wider parabola. The sign also indicates whether reflection across the x-axis occurs.
Converting from standard form uses the axis formula h = -b/(2a). Substituting h into the original equation gives k, completing the vertex coordinates. This method avoids unnecessary expansion and supports accurate graph analysis.
Factored form is especially useful when roots are already known. The vertex lies halfway between two real roots because parabolas are symmetric. Evaluating the function at that midpoint produces the vertical coordinate.
The calculator also connects algebra with parabola geometry. The value p = 1/(4a) locates the focus and directrix. Every point on the parabola remains equally distant from them.
Root behavior depends on the discriminant b² - 4ac. A positive result gives two real roots, zero gives one repeated root, and a negative result gives complex roots. Exact radical notation is shown when appropriate.
Three points with distinct x-values generally determine one vertical quadratic. However, collinear points produce a linear equation, not a parabola. Repeated x-values can also prevent a unique vertical function.
Graph controls help inspect narrow, wide, shifted, or steep curves. Markers show the vertex, intercepts, axis, focus, directrix, and latus rectum. Export tools support classroom, engineering, and study documentation.
Always verify that coefficient a is not zero. A zero value removes the squared term and changes the equation type. Accurate inputs produce reliable algebraic and geometric conclusions every time.
Common Mistakes
- Using h instead of -h inside the parentheses.
- Allowing a to equal zero.
- Forgetting to square the entire x - h expression.
- Using the midpoint of roots as k instead of h.
- Rounding too early before calculating roots or focus.
- Entering three points with repeated x-coordinates.
Frequently Asked Questions
1. What is vertex form?
Vertex form is y = a(x - h)² + k. It displays the vertex directly as (h, k).
2. How is the vertex found from standard form?
Calculate h = -b/(2a), then substitute h into the quadratic to obtain k.
3. Can the calculator accept fractions?
Yes. Inputs such as 1/2, -3/4, and 5/8 are accepted.
4. What happens when a equals zero?
The equation is no longer quadratic, so the calculator reports an error.
5. Why are some roots complex?
Complex roots occur when the discriminant is negative and the graph never crosses the x-axis.
6. How are focus and directrix calculated?
The calculator uses p = 1/(4a), focus (h, k + p), and directrix y = k - p.
7. Can three points define a parabola?
Yes, when their x-values are distinct and they produce a nonzero quadratic coefficient.
8. What does the width classification mean?
It compares |a| with one to determine whether the parabola is narrower, wider, or unchanged.
9. Can I save the results?
Yes. Use copy, CSV, PDF or print, and graph image controls.