# Graphing Quadratic Inequalities

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

## Example 1

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* = *x*^{2} - 2*x* - 3 are *x* = -1, 3.

Solving a quadratic equation by factoring

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

Draw *y* = *x*^{2} - 2*x* - 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

The roots of *y* = *x*^{2} + 3*x* are *x* = -3, 0.

Solving a quadratic equation by factoring

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

Draw *y* = *x*^{2} + 3*x*

using a dashed line.

('<' doesn't include '='.)*y* is lesser than the right side.

So color the lower region of the function.

## Example 3

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 *x*^{2} is (-).*y* is greater than the right side.

So color the upper region of the function.

## Example 4

The vertex of *y* = -*x*^{2} + 4 is (0, 4).

Quadratic function: vertex form

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

Draw *y* = -*x*^{2}

using a solid line.

('≤' includes '='.)

The graph is opened downward

because the coefficient of *x*^{2} is (-).*y* is lesser than the right side.

So color the lower region of the function.