Graph PlottersPlot y = ax² + bx + c and see what a quadratic graph looks like, including its vertex, roots, axis, and direction.
Hover to inspect coordinates. Drag the graph to pan the view or use the wheel and zoom controls to explore the function.
ABOUT THE TOOL
A quadratic function has the form f(x) = ax² + bx + c, where a is not zero. Its graph is a parabola. The sign of a determines whether it opens up or down, while b and c help determine the vertex, axis of symmetry, and y-intercept.
HOW TO USE IT
A quadratic graph calculator is useful because the equation and the parabola answer different questions. In y = ax² + bx + c, the leading coefficient a controls whether the curve opens up or down and how wide it is. The coefficient c is the y-intercept, while b helps determine the location of the axis of symmetry. Graph Plotters keeps those ideas visible with a live curve, a formatted equation, and result cards for the vertex, axis, roots, and discriminant.
The vertex occurs at x = −b/(2a), and its y-coordinate comes from substituting that x-value into the function. The discriminant b² − 4ac tells you how many real roots to expect: positive means two, zero means one repeated root, and negative means no real x-intercepts. Enter large or small coefficients directly when you need them; the graph viewport can be zoomed or fitted without changing the equation.
Use this online quadratic graph calculator for homework, algebra practice, and checking a graph from standard, factored, or vertex form. Try the quick examples, change one coefficient, and move across the curve to read coordinates. If you set a to zero, the tool identifies the lower-degree linear fallback instead of returning a misleading parabola summary.
A quadratic graph is a parabola because its highest power is x². Standard form y = ax² + bx + c is convenient for identifying the y-intercept, factored form y = a(x − r₁)(x − r₂) makes roots visible, and vertex form y = a(x − h)² + k makes the turning point visible. These are not different functions; they are different views of the same relationship. The calculator helps you move between the visual features and the coefficients without losing the original equation.
For y = x² − 4x + 3, the roots are x = 1 and x = 3, so the x-intercepts are (1, 0) and (3, 0). The axis of symmetry lies halfway between them at x = 2, and substituting x = 2 gives the vertex (2, −1). Enter the example and use the markers to check each prediction. Changing a, b, or c then shows which feature moves, helping you understand a quadratic graph rather than only memorizing the quadratic formula.
For y = 2x² − 8x + 6, the axis is x = 2 because −b/(2a) = 8/4. Substitution gives the vertex (2, −2), and factoring gives 2(x − 1)(x − 3), so the roots are 1 and 3. The average of the roots is also 2, confirming the axis of symmetry. Entering the same example in the calculator lets you check the vertex, roots, and discriminant from three connected viewpoints.
The discriminant Δ = b² − 4ac predicts how the parabola meets the x-axis. For y = x² + 2x + 5, Δ = 4 − 20 = −16, so there are no real roots and the upward-opening curve stays above the axis. For y = x² − 6x + 9, Δ = 0, so the graph touches the axis once at x = 3. These quick checks make a graph easier to interpret.
Explore related tools as you study functions: start with the online graphing calculator to plot a general equation, then use the linear graph calculator for constant rates and the slope calculator for rise over run. When a curve meets the x-axis, check the x intercept calculator; for parabolas, compare the quadratic graph calculator and vertex calculator. Use the cubic graph calculator for polynomial end behavior, the sine graph calculator and cosine graph calculator for periodic models, and the phase shift calculator for horizontal movement. The function transformations calculator compares parent and transformed curves, while the absolute value graph calculator explains V-shaped graphs. For growth and inverse relationships, try the exponential growth graph calculator, reciprocal graph calculator, and log graph calculator. For asymptote and periodic-function practice, use the tangent graph calculator, secant graph calculator, and cosecant graph calculator to compare branches, gaps, and repeating behavior.
COMMON QUESTIONS
The x-coordinate is −b/(2a). Substitute that value into the function to find the y-coordinate.
The discriminant b² − 4ac tells you how many real x-intercepts the parabola has: two, one, or none.
It opens upward when a is positive and downward when a is negative.
Calculate Δ = b² − 4ac. A positive discriminant means two distinct real roots, zero means one repeated real root, and a negative value means no real roots.
Changing c changes the y-intercept directly. It can also change the vertex and roots because the whole equation has changed.
A smaller positive |a| makes an upward parabola wider, while a larger |a| makes it narrower. A negative a also reverses the opening direction.