Understanding Absolute Value Quadratic Functions
An absolute value quadratic function applies absolute value to an entire quadratic expression. A common form is y = |ax² + bx + c|. First, the quadratic is evaluated normally. Then every negative output becomes positive. Positive outputs remain unchanged. This transformation keeps the graph on or above the x-axis. It also creates useful symmetry around the quadratic axis.
How the Graph Changes
The original quadratic may pass below the x-axis. Absolute value reflects those sections upward. Real roots stay at the same x-values because zero remains zero. When two distinct roots exist, reflected sections usually create sharp cusp points at those roots. The original vertex may become a local maximum after reflection. If no real roots exist, the quadratic never crosses zero. The transformed graph then behaves like an upward-opening curve after any needed reflection.
Roots, Vertex, and Range
The discriminant determines whether real roots exist. A positive discriminant gives two real roots. A zero discriminant gives one repeated root. A negative discriminant gives no real roots. The vertex uses x = −b/(2a). Its y-value comes from substitution into the quadratic. For the absolute function, outputs never become negative. The range therefore starts at zero whenever real roots exist. Without real roots, the minimum equals the absolute value of the quadratic vertex.
Piecewise Interpretation
Piecewise notation makes the transformation clear. Wherever the quadratic is nonnegative, the function equals ax² + bx + c. Wherever it is negative, the function equals −(ax² + bx + c). This calculator identifies those intervals automatically. It also shows equation solutions, sample values, and graph behavior. These features support algebra study, checking homework, exploring transformations, and comparing exact formulas with decimal results.
Equation and Coefficient Exploration
The equation solver checks both quadratic targets created by the absolute value. This reveals every real solution for a selected nonnegative level. The inverse section also estimates a missing coefficient from a known point. Because absolute value hides the original sign, two coefficient candidates can sometimes satisfy the same condition.