Conversion tool
Enter vertex form values
Use the pattern y = a(x − h)2 + k.
Understanding Vertex and General Forms
Quadratic vertex form describes a parabola through its turning point. The pattern is y = a(x - h)² + k. The point (h, k) is the vertex. The value a controls width and direction. A positive value opens upward. A negative value opens downward. Larger absolute values create narrower curves. Smaller nonzero values create wider curves. This direct comparison makes each coefficient easier to interpret and verify.
Formula Used
The conversion uses expansion and coefficient collection. Start with y = a(x - h)² + k. Expand the square as x² - 2hx + h². Multiply every term by a. This gives ax² - 2ahx + ah² + k. Combine the constant terms. The final general form is y = Ax² + Bx + C. Here, A = a, B = -2ah, and C = ah² + k.
How to Use This Calculator
Enter a, h, and k from the vertex equation. Never enter zero for a, because the graph would stop being quadratic. Choose the desired precision. Select normal or scientific notation. Pick the variable symbol used in the displayed equation. Press the conversion button. The result appears above the form. Review the coefficients before copying the equation.
Additional Quadratic Properties
The calculator also evaluates important quadratic properties. The discriminant is B² - 4AC. A positive discriminant gives two distinct real roots. A zero discriminant gives one repeated real root. A negative discriminant gives two complex roots. The axis of symmetry remains x = h. The y-intercept equals C. The opening direction depends on A.
Managing Signs Correctly
Sign handling deserves attention. When h is positive, the vertex factor contains x minus h. When h is negative, the factor becomes x plus its absolute value. The middle coefficient always follows B = -2ah. Therefore, changing the sign of h changes the sign of B. The constant C includes h squared, so the sign of h disappears inside that square. The value k is then added without alteration.
Precision and Output Choices
Exact symbolic work is ideal for simple integers and fractions. Decimal output is useful for measurements, estimates, and graphing software. Scientific notation helps when coefficients are extremely large or small. Rounding can slightly change displayed roots or the discriminant. The calculation still uses the entered numeric values. Increase precision when close comparisons matter.
Checking the Conversion
A manual check can prevent mistakes. Substitute x = h into the converted general equation. The result should equal k. Compare the leading coefficient with a. It must remain unchanged. Next, compute -B divided by 2A. The answer should equal h. These checks verify the vertex, scale, and axis. They are valuable when working with negative numbers or many decimals.
Practical Uses
This conversion supports graphing, equation solving, optimization, and classroom review. General form exposes coefficients directly. Vertex form exposes geometric structure. Moving between both forms builds stronger algebra understanding. Use the steps panel to study each expansion. Use the analysis panel to inspect roots and direction. Then save the final equation for later work.
Common questions
Frequently Asked Questions
What is quadratic vertex form?
Vertex form is y = a(x − h)² + k. It shows the vertex directly as (h, k). The coefficient a controls the parabola’s direction and width.
What is general form?
General form is y = Ax² + Bx + C. It displays the quadratic, linear, and constant coefficients directly. This form is useful for discriminants, intercepts, and many algebraic methods.
Which formulas perform the conversion?
Use A = a, B = −2ah, and C = ah² + k. These formulas come from expanding the squared binomial and collecting like terms.
Can coefficient a equal zero?
No. When a equals zero, the squared term disappears. The expression becomes constant instead of quadratic, so vertex-form quadratic rules no longer apply.
How are negative h values entered?
Enter the signed value of h. For example, y = 2(x + 3)² − 1 has h = −3 because x + 3 equals x − (−3).
Why does the middle coefficient change sign?
The middle coefficient is B = −2ah. Its sign depends on both a and h. A sign change in either value can reverse the sign of B.
What does the discriminant show?
The discriminant is B² − 4AC. A positive result gives two real roots. Zero gives one repeated root. A negative result gives two complex roots.
Does conversion change the vertex?
No. Both equations describe the same parabola. Only the algebraic arrangement changes. The vertex remains (h, k), and the axis remains x = h.
Why are rounded answers sometimes different?
Displayed values follow the selected precision. Heavy rounding can slightly alter visible roots or discriminants. Choose more decimal places when values are close or detailed comparison matters.
Can the calculator handle decimal values?
Yes. It accepts positive, negative, integer, decimal, and scientific numeric input. You can also choose trimmed, fixed, or scientific output formatting.
How can I verify the result manually?
Check that A equals a. Then confirm −B divided by 2A equals h. Finally, substitute x = h into the general equation. The result should equal k.