Graphing Quadratic Inequalities

Visualize quadratic inequalities and their solutions on a 2D plane.

Enter the coefficients of the quadratic expression and select the inequality type to generate a detailed analysis and graph description.

Examples

Click on any example to load it into the calculator.

Standard Upward Parabola

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A simple inequality with a parabola opening upwards and two distinct roots.

y > 1x² + -4x + 3

Downward Parabola (Solid Line)

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An inequality with a parabola opening downwards and a non-strict inequality.

y ≤ -1x² + 2x + 3

No Real Roots

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An inequality where the parabola does not intersect the x-axis.

y < 2x² + 3x + 4

Vertex on X-Axis

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A perfect square quadratic where the vertex is the only root.

y ≥ 1x² + -6x + 9

Other Titles
Understanding Graphing Quadratic Inequalities: A Comprehensive Guide
Learn how to graph quadratic inequalities, interpret the results, and understand their applications in various fields.

What are Quadratic Inequalities? Core Concepts

  • Understanding the Parabola
  • The Role of the Inequality Sign
  • Solid vs. Dashed Lines and Shaded Regions
A quadratic inequality is a mathematical statement that relates a quadratic expression to a value using an inequality symbol such as >, <, ≥, or ≤. When graphed on a two-dimensional Cartesian plane, it involves a parabola, which is the characteristic U-shape of a quadratic function.
The Parabola: y = ax² + bx + c
The core of the inequality is the quadratic function y = ax² + bx + c. The coefficient 'a' determines the direction the parabola opens: upwards if 'a' is positive, and downwards if 'a' is negative. The vertex is the minimum (if opening up) or maximum (if opening down) point of the parabola.
Interpreting the Inequality
The inequality sign determines which part of the plane is the solution set. For inequalities involving y > ... or y ≥ ..., the solution region is above the parabola. For y < ... or y ≤ ..., the solution is below the parabola. The line of the parabola itself is solid for ≥ and ≤ (indicating points on the line are included in the solution) and dashed for > and < (indicating points on theline are not included).

Key Concepts Illustrated

  • y > x² - 1: A dashed parabola opening upwards, with the region *above* the curve shaded.
  • y ≤ -x² + 4: A solid parabola opening downwards, with the region *below* the curve shaded.

Step-by-Step Guide to Using the Calculator

  • Entering the Coefficients
  • Selecting the Inequality Type
  • Interpreting the Results Section
Our calculator simplifies the process of graphing quadratic inequalities into a few easy steps. Follow this guide to get accurate results quickly.
Inputting Your Inequality
1. Coefficient 'a': Enter the coefficient of the x² term. Remember, this value cannot be zero.
2. Coefficient 'b': Enter the coefficient of the x term.
3. Coefficient 'c': Enter the constant term.
4. Inequality: Choose the correct symbol (>, <, ≥, or ≤) from the dropdown menu to define the relationship.
Analyzing the Output
After clicking 'Graph Inequality', the calculator provides a comprehensive breakdown:
  • Vertex, Roots, Focus: These are the key geometric properties of the parabola.
  • Graph Description: This sentence summarizes the visual representation of the inequality, including the parabola's line style (solid/dashed) and the shaded solution region (above/below).

Practical Walkthrough

  • For y ≥ 2x² - 3x + 1, enter a=2, b=-3, c=1 and select '≥'.
  • The calculator will show it opens upwards, has a solid line, and is shaded above.

Real-World Applications of Quadratic Inequalities

  • Physics and Engineering
  • Business and Economics
  • Optimization Problems
Quadratic inequalities are not just abstract mathematical concepts; they have practical applications in various fields.
Projectile Motion
In physics, the height of a projectile over time can be modeled by a quadratic function. An inequality can be used to determine the time intervals during which the object is above a certain height. For example, finding when a ball thrown in the air is at least 10 feet off the ground.
Profit Maximization
In economics, revenue and cost functions are often quadratic. A business might use a quadratic inequality to determine the range of production levels or prices that will guarantee a profit above a certain threshold.
Design and Architecture
Architects designing structures with parabolic arches, like bridges or ceilings, might use inequalities to ensure the dimensions meet certain constraints, such as being below a maximum height or enclosing a minimum area.

Application Examples

  • Finding when a rocket's altitude, h(t) = -16t² + 100t, is greater than 50 feet.
  • Determining the price range for a product to ensure profit P(x) = -5x² + 200x - 1000 is at least $500.

Mathematical Derivations and Formulas

  • Finding the Vertex
  • The Quadratic Formula for Roots
  • Calculating the Focus and Directrix
The results provided by the calculator are based on fundamental formulas from algebra and geometry.
Vertex Formula
The vertex of a parabola y = ax² + bx + c is a point (h, k). The x-coordinate, h, is found using the formula h = -b / (2a). The y-coordinate, k, is found by substituting h back into the quadratic equation: k = a(h)² + b(h) + c.
Quadratic Formula
The roots (x-intercepts) are the points where the parabola crosses the x-axis (where y=0). They are calculated using the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / (2a). The term inside the square root, b² - 4ac, is called the discriminant. If it's positive, there are two distinct roots. If it's zero, there is one root (the vertex is on the x-axis). If it's negative, there are no real roots.
Focus and Directrix
For a parabola, the focus is a point and the directrix is a line. Every point on the parabola is equidistant from the focus and the directrix. If the vertex is (h, k), the focus is (h, k + p) and the directrix is y = k - p, where p = 1 / (4a).

Core Formulas

  • For y = 2x² - 8x + 6: Vertex x = -(-8)/(2*2) = 2. Vertex y = 2(2)² - 8(2) + 6 = -2. So, Vertex is (2, -2).
  • Roots: x = [8 ± sqrt((-8)² - 4*2*6)] / (2*2) = [8 ± sqrt(16)] / 4 = (8 ± 4)/4. Roots are x=3 and x=1.

Common Misconceptions and Correct Interpretation

  • Inequality vs. Equation
  • The Meaning of Shading
  • Boundary Line Errors
Understanding quadratic inequalities requires avoiding common pitfalls related to their interpretation.
A Solution is a Region, Not a Number
A common mistake is to think the solution is a single number or a pair of numbers. While solving a quadratic equation yields specific x-values (the roots), solving a quadratic inequality on a 2D plane yields an entire region of (x, y) coordinate pairs.
Dashed vs. Solid Lines
Forgetting to distinguish between strict (<, >) and non-strict (≤, ≥) inequalities is another error. A dashed line signifies that the parabola itself is a boundary and not part of the solution. A solid line means the points on the parabola are included in the solution set.
Shading the Wrong Side
It's easy to get confused about whether to shade above or below the parabola. A simple trick is to test a point, like (0,0). If plugging x=0, y=0 into the inequality results in a true statement, then the region containing the origin is the solution. If not, the other side is the solution. The calculator automates this, but it's a useful concept for manual graphing.

Points to Remember

  • For y > x²: Test (0,1). Is 1 > 0²? Yes. So shade the region containing (0,1), which is *inside* the upward-opening parabola (i.e., above it).
  • For y < x² - 1: Test (0,0). Is 0 < 0² - 1? No, 0 is not less than -1. So shade the region that does *not* contain the origin, which is *outside* the parabola (i.e., below it).