range of x^2-4x+3 — step-by-step solution

Let's find the real values taken by

x^{2} - 4 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 = -4

⇥

\frac{4}{2}

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

2

, their greatest common divisor:

\frac{4}{2}

2

⇥

2

⇥

2^{2} - 4 \cdot 2 + 3

⇥ⓘ Powers are computed before multiplications, additions and subtractions. An integer power is a repeated product:

2^{2}

2 \cdot 2

4

■Multiplications are done before additions and subtractions:

4 \cdot 2

8

⇥

4 - 8 + 3

⇥ⓘ Start from

4

and move back

8

on the number line, going below zero:

4 - 8

-4

⇥

- 4 + 3

⇥ⓘ Start from

-4

and move forward

3

on the number line, getting closer to zero:

- 4 + 3

-1

⇥

-1

The vertex is at

x

2

, with value

-1

The square coefficient is positive: this value is the minimum, and the function is unbounded above

▶Range:

[-1, +∞)