QEVT

Math & Statistics

Quadratic Equation Value Table Calculator

Generate a complete x-y value table for a quadratic function, including the vertex, discriminant, row increment, endpoint values, derivative slope, and comparison gap.

Coefficient aCoefficient bCoefficient cStarting xEnding xValue-table rowsEntered y comparison

Vertex x-coordinate-

Vertex y-coordinate-

Discriminant-

Even x increment-

y at starting x-

y at ending x-

Ending y minus starting y-

Vertex y minus entered comparison-

Decision view

Quadratic curve and sampled value-table rows

Quadratic curve and sampled value-table rowsThe continuous parabola, sampled rows, vertex, and comparison level share one coordinate view.

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

Coefficient a	Vertex x-coordinate	Vertex y-coordinate	Discriminant	Even x increment	y at starting x	y at ending x	Ending y minus starting y	Vertex y minus entered comparison

Period-by-period detail

Complete quadratic x-y value table

Each evenly spaced x value recalculates y, derivative slope, and distance from the vertex.

How to use Quadratic Equation Value Table Calculator

Enter coefficients a, b, and c.
Choose the starting x, ending x, and number of rows.
Enter an optional y comparison.
Read the curve and the complete x-y-slope table together.

Calculator guide

Understanding Quadratic Equation Value Table Calculator

A quadratic value table samples one continuous parabola at evenly spaced x-coordinates. The table is most useful when the interval, increment, vertex, and endpoint values are interpreted together.

Endpoints included n rows create n-1 gaps.

Vertex is analytical It is not limited to a sampled row.

Slope changes linearly The derivative is 2ax+b.

Table and curve agree Both use identical inputs.

Calculation method

How the calculation works

Construct the quadratic equation value table across the exact entered x interval, recalculating y, derivative slope, and distance from the vertex for every row. Calculate the vertex and row increment first, then evaluate y = ax² + bx + c and its derivative at every evenly spaced x-coordinate.

Detailed calculation process

Build an evenly spaced quadratic value table

The default function is y = 1.5x² - 4x + 2, sampled from x = -3 to x = 7 in 21 rows and compared with y = 10.

General formula: x_v = -b/(2a)y_v = ax_v^2+bx_v+cD = b^2-4acDelta_x = (x_1-x_0)/(n-1)x_j = x_0+(j-1)Delta_xy_j = ax_j^2+bx_j+cm_j = 2ax_j+b The vertex locates the turning point, the discriminant describes real-root structure, and the interval is divided into n minus one equal gaps so that both endpoints appear in the table.

What each symbol means

a, b, c Quadratic coefficients (dimensionless).

x_v, y_v Vertex coordinates (x-units and y-units).

D Discriminant (coefficient-squared units).

x_0, x_1 Entered starting and ending x-values.

n, j Row count and one-based row index.

Delta_x Even x increment per row (x-units).

x_j, y_j, m_j Row x, function value, and derivative slope.

Worked substitution with the default inputs

1. Locate the vertex x_v = -(-4)/(2×1.5) = 1.333333y_v = 1.5(1.333333)^2-4(1.333333)+2 = -0.666667 The axis of symmetry lies inside the selected interval.

2. Check the discriminant D = (-4)^2-4(1.5)(2) = 16-12 = 4 A positive discriminant means the parabola has two real x-axis intersections.

3. Divide the interval Delta_x = [7-(-3)]/(21-1) = 10/20 = 0.5 Twenty equal gaps create 21 rows including both endpoints.

4. Evaluate the endpoints y(-3) = 27.5y(7) = 47.5y-span = 47.5-27.5 = 20 Endpoint values do not describe the minimum, which occurs at the vertex.

5. Reconcile the comparison y_v-10 = -0.666667-10 = -10.666667 The vertex is 10.666667 y-units below the entered comparison level.

The default table advances by 0.5, captures the vertex near (1.333, -0.667), and runs from y = 27.5 to y = 47.5 at its two endpoints.

Mathematical view

See the continuous curve behind the table

The plot connects the exact quadratic curve, highlights every sampled x-row, and marks the vertex and entered comparison level.

Curve Continuous function.

Sample dots Rows in the table.

Vertex Analytical turning point.

Comparison Entered horizontal reference.

Worked situations

Practical examples

Twenty-one rows from -3 to 7 create a 0.5 increment.
The positive discriminant of 4 indicates two real roots.
The vertex is lower than both endpoint values.

Better inputs

Useful tips

Use more rows when curvature between samples matters.
Choose an interval that contains the vertex or roots of interest.
Do not infer a continuous maximum from endpoints alone.

Before relying on the result

Limitations and common mistakes

The value table samples rather than exhausts the continuous function.
Very large coefficients or intervals can create large displayed values.
The derivative column is analytical and does not estimate slope from neighboring rows.

Reference

Key terms

Vertex: Turning point of the parabola.
Discriminant: b² - 4ac, which controls real-root count.
Increment: Equal spacing between adjacent x rows.

Important note

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

Frequently asked questions

Why are there 20 gaps for 21 rows?

Both endpoints occupy rows, leaving one fewer interval between them.

Can the vertex fall between table rows?

Yes. It is calculated analytically and plotted independently.

What does a positive discriminant mean?

The quadratic has two distinct real roots.

Does the table approximate y?

No. Each listed y is evaluated directly from the quadratic formula.