How to draw quadratic inequality graph?

To draw a quadratic inequality graph, treat the inequality as a function and follow these steps:

Steps to Graphing a Quadratic Inequality

1. Treat the inequality as a function: Replace the inequality sign (>, <, ≥, ≤) with an equal sign (=). This will allow you to graph the corresponding quadratic equation.

2. Find the Vertex: The vertex is a crucial point for graphing the parabola. You can find it by completing the square or using the formula x = -b/2a (where the quadratic is in the form ax² + bx + c). Plug this x-value back into the equation to find the corresponding y-value.

3. Create a Table of Values: Make a table of x and y values around the vertex. This helps you plot several points to accurately draw the parabola. Choose x-values both smaller and larger than the x-value of the vertex.

4. Graph the Parabola: Plot the points from your table and draw a smooth curve (parabola) through them.

   • If the inequality is strict ( < or > ), use a dashed line to indicate that the points on the parabola are not included in the solution.
   • If the inequality includes "equal to" (≤ or ≥), use a solid line to indicate the points on the parabola are included in the solution.

5. Shade the Correct Region: Choose a test point not on the parabola (e.g., (0,0) if it's not on the parabola).

   • Substitute the test point's coordinates into the original inequality.
   • If the inequality is true, shade the region containing the test point.
   • If the inequality is false, shade the region not containing the test point.

Example

Let's graph the inequality y > x² - 2x - 3

1. Treat as a function: y = x² - 2x - 3

2. Find the vertex:

   • x = -(-2) / (2 * 1) = 1
   • y = (1)² - 2(1) - 3 = -4
   • Vertex: (1, -4)

3. Table of Values:

   x	y
   -1	0
   0	-3
   1	-4
   2	-3
   3	0

4. Graph the Parabola: Draw a dashed parabola through these points, since the inequality is '>'.

5. Shade the Region: Test the point (0,0):

   • 0 > (0)² - 2(0) - 3 => 0 > -3 (True)
   • Shade the region above the parabola because (0,0) makes the inequality true.

By following these steps, you can accurately graph any quadratic inequality. Remember the key is to treat the inequality as a function initially, identify the vertex, plot points, and then determine the correct region to shade.