# Graphing Quadratic Inequalities

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

## Example 1: Graph *y* ≥ *x*^{2} - 2*x* - 3

Set (right side) = 0.

And factor the right side.

Factor a quadratic trinomial

Find a pair of numbers

whose product is the constant term [-3]

and whose sum is the middle term's coefficient [-2].

The constant term is (-).

So the signs of the numbers are different:

one is (+), and the other is (-).

(-1, 3) are not the right numbers.

[-3] = -3⋅1

-3 + 1 = [-2]

So -3 and 1 are the right numbers.

Use -3 and +1

to write a factored form:

(*x* - 3)(*x* + 1) = 0.

Solve (*x* - 3)(*x* + 1) = 0.

1) *x* - 3 = 0

So *x* = 3.

2) *x* + 1 = 0

So *x* = -1.

So the zeros are *x* = 3, -1.

Solving a quadratic equation by factoring

To draw *y* ≥ (*x* - 3)(*x* + 1),

first draw *y* = (*x* - 3)(*x* + 1)

on a coordinate plane.

The inequality sign includes [=].

So draw the function using a solid line.

The coefficient of *x*^{2} is (+).

The zeros are -1 and 3.

So draw a parabola that is opened upward

and that passes through -1 and 3.

Quadratic function - Opened upward, downward

Quadratic function - Finding zeros

See *y* ≥ (*x* - 3)(*x* + 1).

This shows that*y* is [greater than] (or equal to) the right side.

So color the [upper region] of the graph.

## Example 2: Graph *y* < *x*^{2} + 3*x*

Set (right side) = 0.

And factor the right side.*x*^{2} + 3*x* = *x*(*x* + 3)

Common monomial factor

Solve *x*(*x* + 3) = 0.

1) *x* = 0

2) *x* + 3 = 0

So *x* = -3.

So the zeros are *x* = 0, -3.

To draw *y* < *x*(*x* + 3),

first draw *y* = *x*(*x* + 3)

on a coordinate plane.

The inequality sign does not include [=].

So draw the function using a dashed line.

The coefficient of *x*^{2} is (+).

The zeros are -3 and 0.

So draw a parabola that is opened upward

and that passes through -3 and 0.

See *y* < *x*(*x* + 3).

This shows that*y* is [less than] the right side.

So color the [lower region] of the graph.

## Example 3: Graph *y* > -*x*^{2} + 2*x* - 1

-*x*^{2} + 2*x* - 1 = -(*x*^{2} - 2*x* + 1)

-2*x* = -2⋅1⋅*x*

+1 = +1^{2}

(*x*^{2} - 2⋅1⋅*x* + 1^{2}) = (*x* - 1)^{2}

Factor a perfect square trinomial

To draw *y* > -(*x* - 1)^{2},

first draw *y* = -(*x* - 1)^{2}

on a coordinate plane.

The inequality sign does not include [=].

So draw the function using a dashed line.

The coefficient of *x*^{2} is (-).

The vertex is (1, 0)

So draw a parabola that is opened downward

and whose vertex is (1, 0).

Quadratic function: vertex form

See *y* > -(*x* - 1)^{2}.

This shows that*y* is [greater than] the right side.

So color the [upper region] of the graph.

## Example 4: Graph *y* ≤ -*x*^{2} + 4

To draw *y* ≤ -*x*^{2} + 4,

first draw *y* = -*x*^{2} + 4

on a coordinate plane.

The inequality sign does include [=].

So draw the function using a solid line.

The coefficient of *x*^{2} is (-).

The vertex is (0, 4)

So draw a parabola that is opened downward

and whose vertex is (0, 4).

See *y* ≤ -*x*^{2} + 4.

This shows that*y* is [less than] (or equal to) the right side.

So color the [lower region] of the graph.