Graphing Quadratic Inequalities

Graphing Quadratic Inequalities

How to graph quadratic inequalities on a coordinate plane: examples and their solutions.

Example 1

Graph the given inequality on the coordinate plane. y >= x^2 - 2x - 3

To draw the quadratic inequality,
you can either use the vertex of the graph
or use the roots of the graph.

In this example, let's use the roots.

The roots of y = x2 - 2x - 3 are x = -1, 3.

Solving a quadratic equation by factoring

Draw full circles on the roots: x = -1, 3.

Draw y = x2 - 2x - 3
using a solid line.
('≥' includes '='.)

y is greater than (or equal to) the right side.
So color the upper region of the function.

Example 2

Graph the given inequality on the coordinate plane. y < x^2 + 3x

The roots of y = x2 + 3x are x = -3, 0.

Solving a quadratic equation by factoring

Draw empty circles on the roots: x = -3, 0.

Draw y = x2 + 3x
using a dashed line.
('<' doesn't include '='.)

y is lesser than the right side.
So color the lower region of the function.

Example 3

Graph the given inequality on the coordinate plane. y > -x^2 + 2x - 1

The vertex of y = -(x - 1)2 is (1, 0).

Quadratic function: vertex form

Draw an empty circle on the vertex: (1, 0).

Draw y = -(x - 1)2
using a dashed line.
('>' doesn't include '='.)

The graph is opened downward
because the coefficient of x2 is (-).

y is greater than the right side.
So color the upper region of the function.

Example 4

Graph the given inequality on the coordinate plane. y <= -x^2 + 4

The vertex of y = -x2 + 4 is (0, 4).

Quadratic function: vertex form

Draw a full circle on the vertex: (0, 4).

Draw y = -x2
using a solid line.
('≤' includes '='.)

The graph is opened downward
because the coefficient of x2 is (-).

y is lesser than the right side.
So color the lower region of the function.