Graphing Quadratic Inequalities Calculator
Graph quadratic inequalities, draw the parabola boundary, and shade the solution region for y compared with ax^2 + bx + c.
Quadratic inequality graphing tool
This graphing quadratic inequalities calculator plots inequalities such as y < ax^2 + bx + c or y >= ax^2 + bx + c. It shows the boundary parabola, shaded solution region, vertex, axis of symmetry, intercepts, and whether a test point is part of the solution.
A graphing quadratic inequalities calculator solves and graphs inequalities like y < x^2 + 2x - 3. It shows the parabola, boundary line style, shaded solution region, intercepts, and vertex. Use it to check hand solutions and visualize where ordered pairs satisfy the inequality.
Use a dashed boundary for strict inequalities, use a solid boundary for inclusive inequalities, then shade above or below the parabola depending on the inequality symbol.
Graph result
Boundary is -- and the solution is shaded --.
Vertex
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--
Opens
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x-intercepts
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Test point (0,0)
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One-variable interval solution
Interval notation
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Comparison line
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Intersections
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Graphing note: Points in the shaded region satisfy the inequality. Points on a solid boundary are included; points on a dashed boundary are not included.
How to use the graphing quadratic inequalities calculator
- Enter the quadratic: Use coefficients a, b, and c for f(x) = ax^2 + bx + c.
- Choose the inequality symbol: Select y <, y <=, y >, or y >= compared with f(x).
- Add an interval comparison if needed: Use the f(x) compared with d fields to solve the one-variable inequality and draw the horizontal line y = d.
- Adjust the graph window: Set x and y minimums and maximums so the vertex, intercepts, and shaded region are visible.
- Read the boundary: A solid parabola means points on the boundary are included. A dashed parabola means boundary points are excluded.
- Read the shading: Shade above the parabola for greater-than inequalities and below the parabola for less-than inequalities.
Two useful outputs: shaded region and interval notation
Many homework problems use one-variable inequalities like ax^2 + bx + c > d, while many graphing problems use two-variable inequalities like y > ax^2 + bx + c. This calculator shows both views so you can connect the algebra answer to the graph.
Coordinate-plane region
The y compared with f(x) controls the shaded area above or below the parabola. This is the full two-variable solution set.
Number-line interval
The f(x) compared with d field solves where the parabola is above, below, or touching the horizontal line y = d.
Graphing rules for quadratic inequalities
A quadratic inequality in two variables compares y with a quadratic expression. The boundary is the parabola y = ax^2 + bx + c, and the solution is a region of the coordinate plane.
If the inequality is y < f(x) or y <= f(x), shade below the parabola. If the inequality is y > f(x) or y >= f(x), shade above the parabola.
Boundary: y = ax^2 + bx + c
Vertex: x = -b / (2a), y = f(-b / (2a))
Discriminant: b^2 - 4ac
Use a test point when you are graphing by hand. Pick a point not on the boundary, substitute it into the inequality, and shade the side that makes the statement true.
Graphing reference: Khan Academy - quadratic inequalities graphical approach.
Convert the inequality into standard form
| Starting form | Example | How to enter it | Useful clue |
|---|---|---|---|
| Standard form | y <= 2x^2 - 5x + 1 | a = 2, b = -5, c = 1 | Fastest form for this calculator because the coefficients are already visible. |
| Vertex form | y > (x - 3)^2 - 4 | Expand to y > x^2 - 6x + 5, so a = 1, b = -6, c = 5 | The vertex is easy to read before expanding: (3, -4). |
| Factored form | y < -2(x + 1)(x - 4) | Expand to y < -2x^2 + 6x + 8, so a = -2, b = 6, c = 8 | The x-intercepts are easy to read before expanding: -1 and 4. |
Quadratic function reference: OpenStax Algebra and Trigonometry - Quadratic Functions.
Worked example: y < x^2 + 2x - 3
This example shows the same reasoning a student would use in a step-by-step solution. Enter a = 1, b = 2, c = -3, and choose y < f(x).
Step 1
Graph the boundary
Use the boundary equation y = x^2 + 2x - 3. Because the inequality is strict, draw the parabola as a dashed curve.
Step 2
Find key points
The vertex is (-1, -4), the axis of symmetry is x = -1, the x-intercepts are -3 and 1, and the y-intercept is -3.
Step 3
Shade the correct region
Since the inequality is y < f(x), shade below the parabola. Points on the dashed boundary are not part of the solution.
Step 4
Check a test point
At (0,0), the statement becomes 0 < -3, which is false. So the region containing (0,0) should not be shaded.
Ordered-pair testing guide
To check whether a point is in the solution region, substitute its x-value and y-value into the original inequality. This is the safest way to confirm your shading after graphing.
Point inside the shaded region
If the ordered pair makes the inequality true, it belongs to the solution set. The point should appear in the shaded region.
Point outside the shaded region
If the ordered pair makes the inequality false, it is not a solution. The point should appear outside the shaded region.
Point on the boundary
If the boundary is solid, the point can be included. If the boundary is dashed, the point is excluded even if it lies on the parabola.
Boundary and shading reference table
| Inequality | Boundary line | Shade | Boundary included? |
|---|---|---|---|
| y < f(x) | Dashed parabola | Below the parabola | No |
| y <= f(x) | Solid parabola | Below the parabola | Yes |
| y > f(x) | Dashed parabola | Above the parabola | No |
| y >= f(x) | Solid parabola | Above the parabola | Yes |
Inequality-solving reference: Mathematics LibreTexts - Solving Quadratic Inequalities.
What to check before copying the graph
Most graphing mistakes come from using the wrong boundary style, shading the wrong side, or choosing a graph window that hides the important part of the parabola.
Boundary type
Use dashed for strict inequalities and solid for inclusive inequalities.
Shading direction
Greater-than inequalities shade above the parabola; less-than inequalities shade below it.
Graph window
If the graph looks empty, widen the x-range or y-range until the vertex and boundary are visible.
Interesting fact
The graph of a quadratic inequality is not just a curve; it is an entire shaded region. The boundary parabola separates points that satisfy the inequality from points that do not. That is why changing only the inequality symbol can keep the same parabola but flip the shaded solution region.
Frequently Asked Questions
What is a graphing quadratic inequalities calculator?
A graphing quadratic inequalities calculator is a math tool that plots a quadratic boundary such as y = ax^2 + bx + c and shades the region that satisfies the inequality. It helps a student see the solution set in the coordinate plane, connect the equation to its graph, and check whether ordered pairs work.
When is the parabola boundary line dashed or solid?
The parabola boundary line is dashed for y < f(x) and y > f(x) because boundary points are not included in the solution. It is solid for y <= f(x) and y >= f(x) because the boundary is part of the shaded region. This rule works whether the quadratic is written in standard form, vertex form, or factored form.
How do I know which side of the parabola to shade?
For y greater than the quadratic, shade above the parabola. For y less than the quadratic, shade below the parabola. In a step-by-step solution, you can test a point such as (0,0), substitute the x and y variables into the inequality, and shade the region that makes the algebra statement true.
What do the vertex and axis of symmetry tell me?
The vertex is the turning point of the parabola, and the axis of symmetry is the vertical line through that point. If a is positive, the vertex is the minimum point of the boundary; if a is negative, it is the maximum point. The vertex also helps describe the range of the quadratic equation, while the domain is usually all real x-values.
Why are the x-intercepts and y-intercept useful?
The x-intercepts show where the boundary parabola crosses the x-axis, and the y-intercept shows where it crosses the y-axis. Intercepts can help you set a better graph window, compare standard form with factored form, and understand where the parabola is above or below y = 0.
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Disclaimer: This graphing quadratic inequalities calculator is for general educational and informational use only. It provides a visual and numerical estimate based on user-entered coefficients and graph window settings. It may not match every classroom graphing convention, teacher requirement, textbook notation, or assessment format. Always check the inequality symbol, boundary style, graph scale, and final answer format required for your course, homework, test, or project.
Last updated: May 18, 2026