Convert Your Quadratic Function
Choose a source form, enter three values, and calculate the matching standard coefficients.
Example Data Table
| Starting form | Given values | Standard form |
|---|---|---|
| Vertex | a = 2, h = 3, k = −5 | f(x) = 2x2 − 12x + 13 |
| Factored | a = −3, r₁ = −2, r₂ = 4 | f(x) = −3x2 + 6x + 24 |
| Vertex | a = 0.5, h = −4, k = 1.5 | f(x) = 0.5x2 + 4x + 9.5 |
Formula Used
Vertex Form Conversion
b = −2ah and c = ah2 + k.
Factored Form Conversion
b = −a(r1 + r2) and c = ar1r2.
The calculator applies the selected formula and preserves the leading coefficient. It then calculates the axis, vertex, discriminant, roots, and opening direction.
How to Use This Calculator
- Select vertex form or factored form from the first field.
- Enter a nonzero leading coefficient in the a field.
- Enter h and k for vertex form, or r₁ and r₂ for factored form.
- Select Calculate Standard Form to place the result above the form.
- Review coefficients, steps, graph details, and available exports.
Understanding Standard Form
Why Standard Form Matters
A quadratic function describes a curved relationship. Its graph is a parabola. Standard form writes the function as f(x) = ax² + bx + c. Each coefficient has a useful job. The value of a controls opening and width. A positive value opens upward. A negative value opens downward. The value of b helps locate the axis of symmetry. The value of c gives the y-intercept. This format is useful for graphing, comparing functions, and applying formulas. It also lets you calculate the discriminant directly. The discriminant helps identify real roots. Standard form gives one clear structure for many algebra tasks. Teachers often request it because coefficient comparisons become direct and consistent across related problems.
Converting Vertex Form
Vertex form is f(x) = a(x − h)² + k. Start by expanding the squared binomial. The expansion is x² − 2hx + h². Multiply every term by a. Then add k to the constant part. The result becomes ax² − 2ahx + ah² + k. Therefore, b equals −2ah. The constant c equals ah² + k. The vertex remains (h, k). This method preserves the graph information while producing the needed coefficients. Careful sign handling matters. A negative h inside parentheses changes the middle term. Check the expansion before accepting your answer.
Converting Factored Form
Factored form is f(x) = a(x − r₁)(x − r₂). First multiply the two binomials. Their product is x² − (r₁ + r₂)x + r₁r₂. Next multiply all terms by a. The completed form is ax² − a(r₁ + r₂)x + ar₁r₂. Therefore, b equals −a(r₁ + r₂). The constant c equals ar₁r₂. The roots stay visible in this form. Standard form instead highlights the y-intercept and coefficients. Both forms describe the same parabola. Use factored form when roots are known. Use standard form when coefficient analysis is required.
Checking the Result
A quick check prevents small errors. Substitute x = 0 into both forms. Each expression should produce the same y-value. You can also test x = 1 or another simple value. Compare the results carefully. Inspect the leading coefficient too. It must remain a. For vertex form, calculate b with −2ah. Then calculate c with ah² + k. For factored form, check the root sum and product. The middle coefficient uses the negative root sum. The constant uses the root product. These checks work even with decimals and negative values. They make conversions more reliable.
Using the Calculator Well
Choose the input form first. Enter the leading coefficient a. It cannot be zero. Then enter the remaining values shown by the fields. Select vertex form for a, h, and k. Select factored form for a, r₁, and r₂. Submit the values to view standard form above the inputs. Review the coefficient cards and conversion steps. The calculator also shows the vertex, axis, discriminant, and roots when available. Export results when you need a record. Use exact fractions by entering decimals only when necessary. Round final answers only after checking the original values. Clear standard form makes quadratic relationships easier to understand.
Frequently Asked Questions
What is standard form for a quadratic function?
Standard form is f(x) = ax² + bx + c, where a is not zero. It shows the leading coefficient, middle coefficient, and constant term in one expression.
How do I convert vertex form to standard form?
Expand (x − h)² into x² − 2hx + h². Multiply by a, then add k. This produces coefficients a, −2ah, and ah² + k.
How do I convert factored form to standard form?
Multiply (x − r₁)(x − r₂) first. The result is x² − (r₁ + r₂)x + r₁r₂. Multiply every term by a to finish.
Can the leading coefficient be zero?
No. When a equals zero, the function is no longer quadratic. The graph becomes linear or constant instead of a parabola.
What does the coefficient c represent?
The coefficient c is the y-intercept. It is the function value when x equals zero. On the graph, it marks where the parabola crosses the vertical axis.
Why is the middle coefficient negative in factored form?
The negative sign comes from multiplying the binomials. The x-term is the negative sum of the roots, then multiplied by the leading coefficient a.
Can I use decimal values?
Yes. Enter decimal values for coefficients, vertex values, or roots. Keep enough decimal places during calculation. Round the displayed answer only when your task allows it.
Does standard form show the vertex directly?
Not directly. You can find the vertex using x = −b divided by 2a. Substitute that x-value into the function to find the vertex y-value.
What does the discriminant tell me?
The discriminant is b² − 4ac. A positive value gives two real roots. Zero gives one repeated real root. A negative value gives no real roots.
How can I check my conversion?
Test the original and converted expressions with the same x-value. Both should give identical results. Testing x = 0 is especially useful because it checks the constant term.
Are factored and vertex forms equivalent to standard form?
Yes. They are different representations of the same quadratic function. Each form emphasizes different features, while standard form makes coefficient-based analysis convenient.