Calculator
Example data table
| Starting form | Input values | Standard form |
|---|---|---|
| Vertex form | a = 2, h = 2, k = -2 | f(x) = 2x² - 8x + 6 |
| Factored form | a = 1, r₁ = 2, r₂ = 5 | f(x) = x² - 7x + 10 |
| Three points | (0, 6), (2, -2), (4, 6) | f(x) = 2x² - 8x + 6 |
Formula used
Standard form: f(x) = ax² + bx + c
Vertex to standard: a(x - h)² + k = ax² - 2ahx + ah² + k
Factored to standard: a(x - r₁)(x - r₂) = ax² - a(r₁ + r₂)x + ar₁r₂
Vertex: x = -b / 2a, then y = f(x)
Discriminant: D = b² - 4ac
Roots: x = (-b ± √D) / 2a
How to use this calculator
- Select the input type that matches your quadratic expression.
- Enter the required coefficients, vertex values, roots, or points.
- Add optional x-values for a custom value table.
- Choose the number of decimal places for rounded output.
- Press the calculate button to see the standard form.
- Review the steps, roots, vertex, and intercept.
- Use CSV or PDF export to save the result.
About expressing quadratic functions in standard form
Why standard form matters
A quadratic function is often easiest to compare in standard form, f(x)=ax²+bx+c. This form shows the leading coefficient, linear coefficient, and constant term at once. Each value supports a different algebra task. The leading coefficient controls opening direction and vertical stretch. The constant term gives the y-intercept. The middle coefficient helps locate the axis of symmetry. Because of this, standard form is useful in graphing, factoring, optimization, and checking answers.
What this calculator converts
This calculator handles several common starting forms. You may enter a vertex form equation, a factored form equation, direct coefficients, or three points on the parabola. The tool expands the expression and returns a clean standard form. It also calculates extra details. These include the discriminant, roots, vertex, axis, intercept, and a value table. This makes the page more than a converter. It works as a compact quadratic analyzer.
Reading the result
The standard form result follows f(x)=ax²+bx+c. If a is positive, the parabola opens upward. If a is negative, it opens downward. The discriminant tells how many real roots exist. A positive discriminant gives two real roots. A zero discriminant gives one repeated real root. A negative discriminant gives complex roots. The vertex result shows the turning point. It is the minimum point for an upward parabola. It is the maximum point for a downward parabola.
Use cases in maths
Students use this conversion when completing square work, checking factors, and building graphs. Teachers can use it to prepare quick examples. Tutors can compare several equivalent forms during practice. Engineers and analysts may also model motion, profit, or area with a quadratic curve. A clear standard form keeps later calculations organized.
Best practice
Use exact input values when possible. Decimals can be helpful, but rounded data may create rounded results. For three-point mode, choose points with different x-values. Points sharing the same x-value cannot define a normal quadratic function. Review the generated steps before copying the final equation. The steps help you see how expansion, substitution, and simplification created the answer. The export buttons keep results easy to record, share, and compare during longer lessons or homework reviews across several related questions today.
FAQs
What is standard form for a quadratic function?
Standard form is f(x)=ax²+bx+c. The value a cannot be zero. This form shows the main coefficients directly.
Can this calculator convert vertex form?
Yes. Enter a, h, and k from f(x)=a(x-h)²+k. The calculator expands the square and simplifies the equation.
Can this calculator convert factored form?
Yes. Enter a and both roots. It expands a(x-r₁)(x-r₂) into ax²+bx+c.
Can three points define a quadratic?
Yes, if the three x-values are different. The calculator solves the matching quadratic through those points.
What does the discriminant show?
The discriminant shows root type. Positive means two real roots. Zero means one repeated root. Negative means complex roots.
Why is a not allowed to be zero?
If a equals zero, the expression is linear. A quadratic function must include a nonzero squared term.
What is the vertex of a quadratic?
The vertex is the turning point. It is found with x=-b/2a, then substituting that x-value into the function.
Can I export my results?
Yes. After calculation, use the CSV or PDF button to download the result and value table.