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

## Example 1: Graph y ≥ x2 - 2x - 3

Set (right side) = 0.
And factor the right side.

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.

So draw the function using a solid line.

The coefficient of x2 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

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 < x2 + 3x

Set (right side) = 0.
And factor the right side.

x2 + 3x = 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 x2 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 > -x2 + 2x - 1

-x2 + 2x - 1 = -(x2 - 2x + 1)

-2x = -2⋅1⋅x
+1 = +12

(x2 - 2⋅1⋅x + 12) = (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 x2 is (-).
The vertex is (1, 0)

So draw a parabola that is opened downward
and whose vertex is (1, 0).

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 ≤ -x2 + 4

To draw y ≤ -x2 + 4,
first draw y = -x2 + 4
on a coordinate plane.

The inequality sign does include [=].
So draw the function using a solid line.

The coefficient of x2 is (-).
The vertex is (0, 4)

So draw a parabola that is opened downward
and whose vertex is (0, 4).

See y ≤ -x2 + 4.

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

So color the [lower region] of the graph.