Calculator Inputs
Select a method, enter values, and calculate the equation.
Recent Calculations
Your latest successful results stay in this browser.
Formula Used
Vertex form begins with a shifted squared expression. Expansion reveals the standard coefficients. The conversion preserves the same parabola.
How to Use This Calculator
- Select the input method that matches your known information.
- Enter coefficients, vertex coordinates, or a complete equation.
- Choose decimal precision or fraction-style output.
- Add an x-value for an automatic equality check.
- Press Calculate to view steps, roots, and graph details.
- Use the utility buttons to copy or export results.
Worked Example
Consider the equation y = 2(x − 3)² + 5. Expand the binomial before distributing the coefficient. Then combine all constant terms carefully.
y = 2(x − 3)² + 5 y = 2(x² − 6x + 9) + 5 y = 2x² − 12x + 18 + 5 y = 2x² − 12x + 23
Example Data Table
| a | h | k | Vertex form | Standard form |
|---|---|---|---|---|
| 2 | 3 | 5 | y = 2(x − 3)² + 5 | y = 2x² − 12x + 23 |
| −1 | −2 | 4 | y = −(x + 2)² + 4 | y = −x² − 4x |
| 1 | 0 | −9 | y = x² − 9 | y = x² − 9 |
| 0.5 | 4 | −2 | y = 0.5(x − 4)² − 2 | y = 0.5x² − 4x + 6 |
Common Sign Mistakes
The sign inside the bracket is opposite the vertex coordinate. Thus, (x + 4)² has h = −4. Check this sign before expanding every equation.
The middle term equals −2hx before multiplying by a. Distribute a across every binomial term. Combine k only after completing that distribution.
Frequently Asked Questions
What is vertex form?
Vertex form is y = a(x − h)² + k. It shows the vertex directly as (h, k). The coefficient a controls direction and vertical stretch.
What is standard form?
Standard form is y = ax² + bx + c. It displays the polynomial coefficients directly. The constant c also gives the y-intercept.
How is b calculated?
Use b = −2ah after identifying a and h. Keep the negative sign during multiplication. This prevents the most common conversion error.
How is c calculated?
Use c = ah² + k. Square h before multiplying by a. Add k after that multiplication is finished.
Can a equal zero?
No, because a zero value removes the squared term. The expression then stops being quadratic. This calculator therefore rejects a = 0.
Can I enter fractions?
Yes, values such as 3/4 and −5/2 are accepted. A zero denominator is invalid. Fraction output uses a practical rational approximation.
Why does (x + 3) mean h = −3?
Vertex form uses the pattern (x − h). Writing x + 3 equals x − (−3). Therefore, the vertex x-coordinate is negative three.
Does conversion change the graph?
No, both equations represent exactly the same function. Only the algebraic arrangement changes. The verification tool confirms matching y-values.
What do the roots show?
Roots show where the parabola crosses the x-axis. A negative discriminant produces complex roots. The graph then has no real x-intercepts.