QEG

Math & Statistics

Quadratic Equation Graphing Calculator

Evaluate and graph y = ax^2 + bx + c across an entered x interval, with discriminant, vertex, real-root branches, endpoint values, comparison point, and a complete value table.

Coefficient aCoefficient bCoefficient cMinimum x shownMaximum x shownNumber of value-table pointsEntered comparison x

Discriminant-

Vertex x-coordinate-

Vertex y-coordinate-

y at minimum x-

y at maximum x-

y at entered comparison x-

Value-table x step-

Decision view

Quadratic function curve and algebraic landmarks

Quadratic function curve and algebraic landmarksThe entered polynomial is plotted on mathematical axes with vertex, real roots, and comparison point.

Exact scenario comparisonCoefficient b changes while all other entered assumptions remain constant.

Coefficient b	Discriminant	Vertex x-coordinate	Vertex y-coordinate	y at minimum x	y at maximum x	y at entered comparison x	Value-table x step

Period-by-period detail

Complete quadratic x-y value table

Every x coordinate is evenly spaced across the entered interval and recalculates the quadratic y value.

How to use Quadratic Equation Graphing Calculator

Enter nonzero coefficient a and coefficients b and c.
Set the displayed x minimum, maximum, and point count.
Enter a comparison x value.
Inspect axes, intercepts, vertex, comparison point, and the value table.

Calculator guide

Understanding Quadratic Equation Graphing Calculator

A quadratic graph is defined by its coefficients, but discriminant, vertex, roots, and the displayed x interval explain its shape. This calculator plots the actual function and labels those mathematical features on matching axes.

Function drives graph Every plotted y is evaluated from ax^2+bx+c.

Discriminant classifies Positive gives two real roots.

Vertex anchors shape Its x coordinate is -b/(2a).

Step controls sampling The domain is split into N-1 equal intervals.

Calculation method

How the calculation works

Evaluate a quadratic's discriminant, vertex, endpoints, comparison point, and evenly spaced x-y value table across the exact entered graph interval. Compute the discriminant and vertex algebraically, calculate real-root branches when applicable, and evaluate the polynomial at evenly spaced x values.

Detailed calculation process

Derive the defining points of a quadratic before plotting it

The defaults graph y = x^2 - 6x + 5 from x = -2 to x = 8 with 21 evenly spaced points.

General formula: f(x)=ax^2+bx+c; Delta=b^2-4ac; x_v=-b/(2a); y_v=f(x_v); x_(1,2)=(-b +/- sqrt(Delta))/(2a); h=(x_max-x_min)/(N-1) The vertex comes from the axis of symmetry, the discriminant determines real-root behavior, and the value table samples the same polynomial at a constant x step across the entered domain.

What each symbol means

a, b, c Quadratic, linear, and constant coefficients (entered number units).

Delta Discriminant b^2-4ac (coefficient-derived units).

x_v, y_v Vertex coordinates.

x_1, x_2 Real-root branch coordinates when Delta is nonnegative.

x_min, x_max Entered horizontal graph limits.

N, h Number of table points and constant x increment.

Worked substitution with the default inputs

1. Write the default function: f(x) = 1*x^2 + (-6)*x + 5 = x^2-6x+5 Keeping the signed b coefficient visible prevents a sign error.

2. Calculate the discriminant: Delta = (-6)^2 - 4*1*5 = 36-20 = 16 A positive discriminant means two distinct real roots.

3. Find the vertex: x_v = -(-6)/(2*1) = 3; y_v = 3^2-6*3+5 = -4 Because a is positive, this vertex is the minimum.

4. Find both roots: x_1 = (6+4)/2 = 5; x_2 = (6-4)/2 = 1 Substituting either root into the polynomial returns zero.

5. Build and check the graph interval: h = [8-(-2)]/(21-1) = 0.5; f(-2)=21; f(4)=-3; f(8)=21 The equal endpoint values reflect symmetry around x = 3.

The default parabola opens upward, crosses at x = 1 and x = 5, has vertex (3,-4), and is sampled every 0.5 x unit from -2 through 8.

Coordinate plane

Plot the function with its actual algebraic landmarks

The visual uses mathematical axes rather than generic bars.

Parabola Calculated samples follow the entered coefficients.

Root markers Real x-axis crossings are labeled.

Vertex marker The turning point is explicit.

Comparison marker The entered x is evaluated on the same curve.

Worked situations

Practical examples

The default discriminant is 16.
The roots are 1 and 5.
The vertex is (3,-4), centered between the roots.

Better inputs

Useful tips

Choose an x interval that contains the vertex and expected roots.
Increase point count when the displayed interval is wide.
Use symbolic or complex-number methods when the discriminant is negative.

Before relying on the result

Limitations and common mistakes

Coefficient a must be nonzero for a quadratic.
Negative-discriminant cases have no real x-axis intersections; the legacy numeric result fields clip the square root and must not be read as complex roots.
Finite sampling approximates the drawn curve between table points.

Reference

Key terms

Discriminant: b^2-4ac, which classifies real roots.
Vertex: Turning point of the parabola.
Axis of symmetry: Vertical line x = -b/(2a).

Important note

Calculated directly from the entered values using the displayed formula and rounding settings.

Frequently asked questions

Why are there 20 intervals for 21 points?

Both endpoints are included, so N points create N-1 gaps.

How do I know whether the parabola opens upward?

It opens upward when a is positive and downward when a is negative.

What happens when the discriminant is zero?

Both root branches meet at one repeated real root.

Can the chart show complex roots?

No. Complex roots do not lie on the real x-y plane.