Graphing Quadratic Inequalities Calculator: Visualize Parabola Regions
Table of Contents - Graphing Quadratic Inequalities
- How to Use This Calculator
- Understanding Quadratic Inequalities
- How to Graph Quadratic Inequalities Manually
- Real-World Applications
- Common Mistakes and How to Avoid Them
- Related Topics
- How This Calculator Works
- FAQs
How to Use This Calculator - Graphing Quadratic Inequalities
Enter a quadratic inequality like y is greater than x² - 4x + 3, y is less than or equal to -x² + 2x + 5, or x² - 9 is less than 0.
Click "Graph" to see the parabola and shaded solution region. The calculator graphs the parabola, shows whether the curve is solid or dashed, and shades the region that satisfies the inequality.
The results display the coordinate graph, identify the vertex and x-intercepts, show the shaded region clearly, and provide the solution set description.
Understanding Quadratic Inequalities
A quadratic inequality involves a quadratic expression with an inequality symbol instead of an equals sign. The solutions form a region on the coordinate plane rather than just a curve.
The fundamental concept:
Quadratic inequalities like y is greater than x² + 2x - 3 ask: "For which points (x, y) is the y-coordinate greater than the parabola value at that x?" The answer is a region, not just points on a line.
The boundary curve:
The related equation y = ax² + bx + c forms the boundary. This parabola divides the plane into two regions: one where y is greater than the parabola, another where y is less than.
Solid versus dashed:
For "greater than or equal" or "less than or equal," draw a solid parabola (the boundary is included). For strict inequalities (greater than, less than), draw a dashed parabola (boundary not included).
Shading direction:
For y is greater than f(x), shade above the parabola. For y is less than f(x), shade below. If the parabola opens upward and you're shading above, you're shading the "outside" of the bowl. Shading below means shading "inside" the bowl.
Single-variable form:
An inequality like x² - 4x + 3 is less than 0 asks where the parabola is below the x-axis. This gives an interval of x-values rather than a 2D region.
Test point method:
After graphing the parabola, pick a test point not on the curve. Substitute it into the inequality. If true, shade the region containing that point. If false, shade the opposite region.
Why it matters:
Quadratic inequalities model ranges where relationships hold. "Profit is positive when..." or "height is at least..." translate to quadratic inequality regions.
How to Graph Quadratic Inequalities Manually
Let me show you how to graph different types of quadratic inequalities step by step.
Example 1: Basic y is greater than parabola
Graph: y is greater than x² - 4
Step 1: Graph the boundary y = x² - 4 Vertex: (0, -4) Opens upward x-intercepts: x² - 4 = 0, so x = ±2
Step 2: Determine line style "Greater than" is strict, so use dashed curve
Step 3: Test a point Try (0, 0): Is 0 greater than 0² - 4? Is 0 greater than -4? Yes! ✓
Step 4: Shade Since (0, 0) satisfies the inequality, shade the region containing (0, 0), which is above the parabola.
Result: Dashed parabola opening upward with vertex at (0, -4), shaded above.
Example 2: y is less than or equal
Graph: y is less than or equal to -x² + 4
Step 1: Graph boundary y = -x² + 4 Opens downward (negative leading coefficient) Vertex: (0, 4) (maximum point) x-intercepts: -x² + 4 = 0, x = ±2
Step 2: Line style "Less than or equal" includes boundary, so solid curve
Step 3: Test point Try (0, 0): Is 0 less than or equal to -0² + 4? Is 0 less than or equal to 4? Yes! ✓
Step 4: Shade (0, 0) works, so shade below the parabola (inside the downward-opening bowl).
Result: Solid parabola opening down, shaded below/inside.
Example 3: Single-variable inequality
Solve and graph: x² - 5x + 6 is less than 0
Step 1: Find where x² - 5x + 6 = 0 Factor: (x - 2)(x - 3) = 0 Roots: x = 2 and x = 3
Step 2: Determine where parabola is negative Opens upward (positive leading coefficient) Below x-axis between the roots
Step 3: Graph on number line Solution: 2 is less than x is less than 3
←----○===○----→
2 3
Open circles because strict inequality.
Interval notation: (2, 3)
Example 4: Greater than or equal (single variable)
Solve: x² - 2x - 8 is greater than or equal to 0
Step 1: Find roots x² - 2x - 8 = 0 Factor: (x - 4)(x + 2) = 0 x = 4 or x = -2
Step 2: Parabola opens upward Above or on x-axis outside the roots
Step 3: Solution x is less than or equal to -2 OR x is greater than or equal to 4
←====●---------●====→
-2 4
Closed circles for "or equal"
Interval: (-∞, -2] ∪ [4, ∞)
Example 5: Two-variable with vertex form
Graph: y is less than 2(x - 1)² + 3
Step 1: Identify features Vertex form: y = a(x - h)² + k Vertex: (1, 3) Opens upward (a = 2 is positive)
Step 2: Dashed curve Strict inequality
Step 3: Test (0, 0) Is 0 less than 2(0-1)² + 3? Is 0 less than 2(1) + 3 = 5? Yes! ✓
Step 4: Shade below Shade inside the upward-opening parabola.
Example 6: Downward opening
Graph: y is greater than or equal to -x² + 2x + 3
Step 1: Find vertex x-coordinate: -b/(2a) = -2/(2(-1)) = 1 y-coordinate: -(1)² + 2(1) + 3 = 4 Vertex: (1, 4)
Step 2: x-intercepts -x² + 2x + 3 = 0 Multiply by -1: x² - 2x - 3 = 0 Factor: (x-3)(x+1) = 0 x = 3 or x = -1
Step 3: Solid curve (includes "equal")
Step 4: Test (0, 0) Is 0 greater than or equal to -0² + 2(0) + 3? Is 0 greater than or equal to 3? No ✗
Step 5: Shade opposite region Since (0,0) doesn't work, shade outside. For downward parabola, "outside" is above the curve.
Example 7: Finding solution algebraically first
Solve: x² - x - 12 is greater than 0
Step 1: Factor (x - 4)(x + 3) = 0 Roots: x = 4, x = -3
Step 2: Sign analysis Test intervals:
- x is less than -3: try x = -4: 16 + 4 - 12 = 8 is greater than 0 ✓
- -3 is less than x is less than 4: try x = 0: 0 - 0 - 12 = -12 is not greater than 0 ✗
- x is greater than 4: try x = 5: 25 - 5 - 12 = 8 is greater than 0 ✓
Step 3: Solution x is less than -3 OR x is greater than 4
←====○---------○====→
-3 4
Example 8: No solution case
Solve: x² + 4 is less than 0
Step 1: Analyze x² + 4 is always at least 4 (x² is greater than or equal to 0)
Step 2: Conclusion Never negative. No solution: ∅
Example 9: All real numbers
Solve: -x² - 1 is less than 0
Step 1: Simplify -x² - 1 is always negative (both terms negative)
Step 2: Solution All real numbers: (-∞, ∞)
Example 10: With coefficient
Graph: y is less than or equal to 2x² - 8x + 6
Step 1: Find vertex x = -b/(2a) = 8/(2·2) = 2 y = 2(4) - 8(2) + 6 = -2 Vertex: (2, -2)
Step 2: x-intercepts 2x² - 8x + 6 = 0 Divide by 2: x² - 4x + 3 = 0 Factor: (x-3)(x-1) = 0 x = 1, 3
Step 3: Solid parabola (includes equal)
Step 4: Opens upward, shade below/inside
Real-World Applications
Projectile motion constraints:
A ball's height h = -16t² + 48t. Finding when h is greater than or equal to 32 gives the time interval when the ball is at least 32 feet high.
Profit regions:
Profit P = -x² + 40x - 300 where x is items sold. P is greater than 0 shows the production range yielding profit.
Engineering tolerances:
Stress in a beam follows a parabolic function. Finding where stress is less than the material limit defines safe operating ranges.
Economics equilibrium:
Supply and demand curves can be quadratic. Graphing inequalities shows regions of surplus or shortage.
Physics: motion constraints:
Velocity or acceleration functions that are quadratic. Inequalities define time periods with certain properties.
Safety zones:
Splash zones for fountains, range of projectiles, or coverage areas often follow parabolic boundaries. Inequalities define safe or covered regions.
Optimization problems:
Finding where cost is less than budget, where efficiency is greater than a threshold, or where output meets requirements often involves quadratic inequalities.
Common Mistakes and How to Avoid Them
Mistake 1: Wrong line style
Wrong: Using solid curve for strict inequality like y is greater than x²
Right: Strict (greater than, less than) uses dashed. "Or equal" uses solid.
Why it happens: Forgetting that solid means "included." No equal sign means dashed.
Mistake 2: Shading wrong side
Wrong: For y is greater than x², shading below the parabola
Right: "Greater than" means y-values above the curve. Shade above.
Why it happens: Not understanding what "greater than" means geometrically. Test a point to verify.
Mistake 3: Forgetting to test a point
Wrong: Guessing which side to shade
Right: Always test a point (like origin if it's not on the curve) to verify shading.
Why it happens: Assuming without checking. Testing removes all doubt.
Mistake 4: Sign errors finding roots
Wrong: Factoring x² - 5x + 6 as (x - 2)(x + 3)
Right: (x - 2)(x - 3). Check: -2 times -3 = 6 ✓, -2 + -3 = -5 ✓
Why it happens: Arithmetic errors. Always verify factors by expanding.
Mistake 5: Interval direction errors
Wrong: For x² - 4 is less than 0 with roots ±2, answering x is less than -2 or x is greater than 2
Right: Parabola is negative (below x-axis) between roots: -2 is less than x is less than 2.
Why it happens: Not visualizing the parabola. Sketch it or test values.
Mistake 6: Vertex formula errors
Wrong: For y = x² + 4x + 1, finding vertex x-coordinate as 4/2 = 2
Right: x = -b/(2a) = -4/(2·1) = -2
Why it happens: Forgetting the negative sign in the formula. It's negative b, not just b.
Mistake 7: Empty set not recognized
Wrong: Trying to graph x² + 5 is less than 0
Right: Recognize x² + 5 is always positive. No solution. Write ∅.
Why it happens: Not analyzing before calculating. Check if it's even possible.
Related Topics
- Quadratic Formula Calculator - Find roots
- Graphing Inequalities Calculator - Linear inequalities
How This Calculator Works
Step 1: Parse inequality
Extract quadratic expression
Identify inequality symbol
Determine if two-variable (y involved) or single-variable
Extract coefficients a, b, c from ax² + bx + c
Step 2: Find key features
Calculate vertex: x = -b/(2a), y = f(-b/(2a))
Find x-intercepts by solving ax² + bx + c = 0
Determine if parabola opens up (a is greater than 0) or down (a is less than 0)
Find y-intercept: c
Step 3: Determine boundary style
If less than or greater than (strict): dashed curve
If less than or equal or greater than or equal: solid curve
Step 4: Graph parabola
Plot vertex
Plot x-intercepts
Plot y-intercept
Sketch parabola with appropriate style (solid/dashed)
Step 5: Determine shading
Test point method:
Choose test point (often (0,0) if not on curve)
Substitute into inequality
If true: shade region containing test point
If false: shade opposite region
Or analyze directly:
y is greater than f(x): shade above
y is less than f(x): shade below
Step 6: For single-variable inequalities
Find roots of equation
Test intervals between roots
Identify where inequality is true
Express as interval on number line
Step 7: Display results
Show graph with shaded region
Provide solution description
Give interval notation (for single-variable)
List key points (vertex, intercepts)
FAQs
What's the difference between quadratic equations and inequalities?
Equations have = and solutions are specific points. Inequalities use less than, greater than, etc. and solutions are regions or intervals.
How do I know if the parabola should be solid or dashed?
Solid if the inequality includes "or equal to" (less than or equal, greater than or equal). Dashed for strict inequalities (less than, greater than).
Which side do I shade?
For y is greater than f(x), shade above the parabola. For y is less than f(x), shade below. Always test a point to confirm.
What if the parabola doesn't cross the x-axis?
If it opens upward and stays above, the quadratic is always positive. If it opens downward and stays below, always negative. Solution may be all reals or empty set depending on inequality.
How do I find the vertex?
Use x = -b/(2a) to find x-coordinate, then substitute to find y-coordinate. Or complete the square to get vertex form.
What does it mean to solve x² - 4 is less than 0?
Find x-values where the parabola y = x² - 4 is below the x-axis. Graph it or factor to find the interval.
Can I have no solution?
Yes. x² + 1 is less than 0 has no solution because x² + 1 is always positive.
Can the solution be all real numbers?
Yes. x² + 1 is greater than 0 is true for all real x.
How do I handle negative leading coefficients?
a is less than 0 means the parabola opens downward (∩ shape). Everything else works the same.
What if my test point is on the parabola?
Choose a different test point. The test point must be off the curve to determine which side to shade.
Why does the region have two parts sometimes?
For single-variable inequalities like x² is greater than 4, the parabola is above the x-axis in two separate intervals, so the solution is a union.
How do I graph y is greater than x² + 2x - 3?
Graph the parabola y = x² + 2x - 3 as a dashed curve, then shade above it.
What's the difference between less than and less than or equal?
Less than uses dashed curve and doesn't include the boundary. Less than or equal uses solid curve and includes all points on the parabola.
Can I solve quadratic inequalities algebraically?
Yes, find the roots, determine sign of the quadratic in each interval, and identify where the inequality holds.
How do I check my answer?
Pick a point in your shaded region and substitute into the original inequality. It should make the inequality true.
What if there's only one x-intercept?
The vertex touches the x-axis. The parabola is either always non-negative (opens up) or always non-positive (opens down).
How do I write the solution?
For 2D inequalities, describe the shaded region. For single-variable, use interval notation like (-2, 3) or (-∞, -1] ∪ [5, ∞).
What about systems of quadratic inequalities?
Graph each inequality's region, then find where the regions overlap. The intersection is the solution.
Do I always need to graph?
For single-variable inequalities, you can solve algebraically using sign analysis. Graphing helps visualize but isn't always necessary.
Can fractions be involved?
Yes, coefficients and roots can be fractions. The process is identical, just more careful arithmetic.