Quadratic form

The plotted function is y = ax² + bx + c with a non-zero. Coefficient a controls opening direction and vertical scale, b shifts the axis of symmetry and c is the y-intercept. Decimal and negative coefficients are accepted. The page does not parse brackets, powers or functions typed as text.

Set a close to zero and the parabola can look almost linear inside a small window, but it remains quadratic. If a equals zero exactly, use a line graph because vertex and quadratic root formulas no longer apply.

Viewing window

The graph maps entered x and y ranges to a 500 by 260 SVG viewport. Maximum values must exceed minimum values. A root outside the x window will not appear even though the numerical root result reports it. A y window far wider than the curve can make variation look flat.

Choose a window that includes the vertex, y-intercept and expected roots. Then zoom by narrowing ranges. Changing the window changes appearance, not the function. Record the window when sharing a screenshot so another person can reproduce it.

Interactive plot

The blue polyline samples 161 x positions across the window and joins their screen coordinates. A quadratic is smooth, so this is visually adequate at ordinary scales, but it is still a sampled drawing. Extremely narrow or huge windows can expose floating-point and clipping limitations.

Grey horizontal and vertical lines mark y = 0 and x = 0 when those coordinates fall in or near the mapped viewport. If zero lies outside a range, an axis line can sit outside the visible area. The border remains only a frame, not an axis.

Vertex and axis

The vertex x-coordinate is -b/(2a). Substituting it into the function gives vertex y. The axis of symmetry is the vertical line through that x. If a is positive, the vertex is a minimum; if a is negative, it is a maximum.

For x² − 4x + 3, the vertex is (2, -1) and the axis is x = 2. Values at equal horizontal distances from 2 are equal. That symmetry is a useful graph and arithmetic check.

Discriminant and intercepts

The discriminant b² − 4ac determines real x-intercepts. A positive value gives two crossings, zero gives one touching intercept and negative gives none. The root result uses the quadratic formula and does not depend on whether the intercepts fit inside the current window.

A graph can appear to touch when two close roots are unresolved by the display. Use the discriminant and numerical values for classification. In exact algebra, retain radicals when roots are irrational rather than copying a rounded screen coordinate.

Y-intercept

Setting x to zero leaves y = c, so the y-intercept is (0, c). It is visible only if x = 0 and y = c lie in the selected window. If the plotted curve does not pass through the expected point, check the window and coefficient sign.

The y-intercept is not generally the minimum or a root. In an application, c can represent an initial value when x = 0, but only if zero belongs to the model domain.

Transformations

Writing y = a(x − h)² + k reveals vertex (h, k). Expanding gives b = -2ah and c = ah² + k. Changing h moves the parabola horizontally, k moves it vertically and a changes opening and scale.

Use the reported vertex to rewrite a standard-form quadratic into vertex form. Substitute one additional point to confirm a. This links algebraic coefficients to visible transformations instead of treating the plot as a black box.

Applications and domain

Quadratics model simplified projectile height, area, revenue, cost and optimisation. A graph can show turning points and feasible intervals, but application domains may exclude negative time, length or production. Shade or note the valid x range separately.

A fitted quadratic is not reliable indefinitely. Extrapolation can produce impossible growth or decline. Include units, data range and assumptions with any decision based on the vertex or root.

Graph-reading errors

Pixels and line thickness limit precision. Do not read a root to six decimals from a plot. Use the numerical result or solve algebraically. A truncated axis can exaggerate change, and unequal screen scaling makes geometric angle or width comparisons misleading.

Check labelled ranges before saying a curve is steep or flat. Compare function values at chosen x points. Graphs support reasoning, but scale choices can persuade as well as inform.

A reliable checking routine

First calculate the y-intercept from c and the axis from -b divided by 2a. Next substitute the reported vertex x-coordinate into the original expression and check the displayed y-value. If real roots exist, substitute each rounded root and confirm that the resulting y-value is close to zero within rounding tolerance.

Then compare the signs. A positive a must open upward and place the vertex below or level with every other point; a negative a must open downward. The product of two real roots should equal c divided by a, and their sum should equal -b divided by a. These relationships often expose a copied sign or coefficient error faster than redrawing the curve.

Finally choose a viewing window that includes all calculated features and press calculate again. If a numerical feature remains invisible, inspect the x and y ranges before questioning the algebra. Save the coefficients, window and rounded results together. A screenshot without these inputs cannot be independently reproduced and can conceal a misleading scale.

Accessibility and alternatives

The numerical results communicate key features without requiring colour perception. The blue curve should not be the only source used for an answer. A screen reader may not interpret SVG geometry, so use the vertex, roots, intercept and shape text.

For a data table, evaluate the polynomial at evenly spaced x values and list units. For publication or advanced work, use graphing software with labelled axes, export controls and documented precision. This embedded plot is an exploratory companion.

Rendered quadratic plot

The plot updates when Calculate is pressed. Values outside the chosen y-range are drawn beyond the visible frame, so use the window fields to zoom.

Preserve a reproducible graph

Save the three coefficients and all four window limits with any screenshot. Independently substitute the reported roots, vertex and intercept into the polynomial. A graph is a visual sample within a chosen window; the accompanying algebra is what confirms whether a feature is real, rounded, clipped or outside the visible range.