range of -x^2-2x+3 — step-by-step solution

Let's find the real values taken by

- x^{2} - 2 x + 3

, for real

x

The vertex of a parabola

a x^{2} + b x + c

is at

x = - \frac{b}{2 a}

, here with

a = -1

and

b = -2

⇥

\frac{2}{2 \cdot (-1)}

⇥ⓘ A single minus sign does not cancel out, the product of a negative number by a positive number is negative:

2 \cdot (-1)

-2

⇥

\frac{2}{-2}

⇥ⓘ The sign of a fraction goes in front of the bar, whether it comes from the numerator or the denominator:

\frac{2}{-2}

- \frac{2}{2}

⇥

- \frac{2}{2}

⇥ⓘ We simplify the fraction by dividing the numerator and the denominator by

2

, their greatest common divisor:

- \frac{2}{2}

-1

⇥

-1

⇥

- {(-1)}^{2} - 2 \cdot (-1) + 3

⇥ⓘ An integer power is a repeated product:

- {(-1)}^{2}

- ((-1) \cdot (-1))

-1

■Multiplications are done before additions and subtractions. Two minus signs cancel out, the product of two negative numbers is positive:

-2 \cdot (-1)

2

⇥

- 1 + 2 + 3

⇥ⓘ Start from

-1

and move forward

2

on the number line, going past zero:

- 1 + 2

1

⇥

1 + 3

⇥ⓘ Start from

1

and move forward

3

on the number line:

1 + 3

4

⇥

4

The vertex is at

x

-1

, with value

4

The square coefficient is negative: this value is the maximum, and the function is unbounded below

▶Range:

(-∞, 4]