y=x^2-6x+8 — step-by-step solution

We save the definition:

y = x^{2} - 6 x + 8

Let's study the function

y (x) = x^{2} - 6 x + 8

1) Let's find the domain of

x^{2} - 6 x + 8

ⓘ This expression imposes no restriction on the real variable

Domain of definition:

x \in ℝ

2) Let's study the parity of

x^{2} - 6 x + 8

with respect to

x

The domain is

x \in ℝ

This domain is symmetric about 0: if

x

is in the domain, -

x

is too

Replace

x

by -

x

: y(-

x

) =

{(- x)}^{2} - 6 \cdot (- x) + 8

Let's calculate:

{(- x)}^{2} - 6 \cdot (- x) + 8

ⓘ Powers are computed before multiplications, additions and subtractions. With an even integer exponent, the minus sign in the base disappears; with an odd integer exponent, it moves in front of the power:

{(- x)}^{2}

x^{2}

■Two minus signs cancel out:

-6 \cdot (- x)

6 x

x^{2} + 6 x + 8

y(-

x

) - y(

x

) =

12 x

and y(-

x

) + y(

x

) =

2 x^{2} + 16

are nonzero polynomials: the function is neither even nor odd

Parity: neither even nor odd

3) Differentiability

⇥ⓘ The derivative of a sum is the sum of the derivatives:

\frac{d}{d x} (x^{2} - 6 x + 8)

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 6 x) + 0

⇥

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 6 x) + 0

Derivative:

2 x - 6

4) Let's study the variations of

y (x)

: first find where its derivative

y^{'} (x) = 2 x - 6

is zero

Let's solve in

ℝ

2 x - 6 = 0

ⓘ We add

6

to both sides

2 x = 6

ⓘ Both sides are divided by

2

x = \frac{6}{2}

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

2

, their greatest common divisor:

\frac{6}{2}

3

Solution found:

x = 3

y^{'} (x)

is zero at

x = 3

Built from usual functions,

y^{'} (x)

is continuous where it is defined: on each interval where it is never zero, it keeps the sign of its value at any point

y^{'} (0) = -6

< 0:

y

is decreasing on

(-∞, 3)

y^{'} (4) = 2

> 0:

y

is increasing on

(3, +∞)

Let's calculate the values of

y

at the bounds of the table, or its limits at the infinite or excluded bounds

x

tends to

-∞

y (x)

tends to

+∞

y (3) = 3^{2} - 6 \cdot 3 + 8 = -1

x

tends to

+∞

y (x)

tends to

+∞

Let's draw up the variation table:

$x$	$-∞$		$3$		$+∞$
$y^{'} (x)$		-	$0$	+
$y (x)$	$+∞$	↘	$-1$	↗	$+∞$

y

is decreasing on

(-∞, 3]

and increasing on

[3, +∞)

5) Let's read the range from the variation table

⇥On

ℝ

y

is continuous: it takes every value of

[-1, +∞)

Range:

[-1, +∞)

6) On the horizontal axis, y = 0: solve

x^{2} - 6 x + 8 = 0

Let's solve in

ℝ

x^{2} - 6 x + 8 = 0

Given the quadratic form

a x^{2} + b x + c

, let's compute the discriminant:

Δ = b^{2} - 4 a c

. Here

a = 1

b = -6

and

c = 8

Δ = {(-6)}^{2} - 4 \cdot 1 \cdot 8

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

{(-6)}^{2}

(-6) \cdot (-6)

36

Δ = 36 - 4 \cdot 8

ⓘ Multiplications are done before additions and subtractions:

4 \cdot 8

32

Δ = 36 - 32

ⓘ Start from

36

and move back

32

on the number line:

36 - 32

4

Δ = 4

Since

Δ > 0

, the equation has two roots:

The quadratic formula gives

x_{1}

\frac{- b - \sqrt{Δ}}{2 a}

\frac{- (-6) - \sqrt{4}}{2 \cdot 1}

x_{1} = \frac{6 - \sqrt{4}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

x_{1} = \frac{6 - 2}{2}

ⓘ The numerator and the denominator are computed before the division:

6 - 2

4

x_{1} = \frac{4}{2}

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

2

, their greatest common divisor:

\frac{4}{2}

2

x_{1} = 2

The quadratic formula gives

x_{2}

\frac{- b + \sqrt{Δ}}{2 a}

\frac{- (-6) + \sqrt{4}}{2 \cdot 1}

x_{2} = \frac{6 + \sqrt{4}}{2}

ⓘ The square root of a number is the positive number whose square is that number:

\sqrt{4}

2

;

2^{2}

4

x_{2} = \frac{6 + 2}{2}

ⓘ The numerator and the denominator are computed before the division:

6 + 2

8

x_{2} = \frac{8}{2}

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

2

, their greatest common divisor:

\frac{8}{2}

4

x_{2} = 4

Solutions found:

x = {2, 4}

On the vertical axis, x = 0: calculate f(0)

⇥

0^{2} - 6 \cdot 0 + 8

⇥ⓘ A product with a zero factor is zero:

-6 \cdot 0

0

■Adding 0 changes nothing:

0 + 8

8

⇥

0 + 8

⇥ⓘ Adding 0 changes nothing:

0 + 8

8

⇥

8

Axis intercepts:

{(2, 0), (4, 0), (0, 8)}

7) Let's find the asymptotes of

x^{2} - 6 x + 8

, for real

x

x^{2} - 6 x + 8

is a polynomial of degree

2

: it is defined on ℝ and

\frac{f (x)}{x}

tends to infinity at -∞ and +∞, so no line approaches its graph

So the function has no asymptote

🖼/draw?f1=x%5E2+-+6*x+%2B+8&v=x&b=-10.0&e=10.0&min=-12.0&max=132.0

Asymptotes:

\emptyset

▶Study of $y (x)$ : Domain of definition: $x \in ℝ$ ; Parity: neither even nor odd; Derivative: $2 x - 6$ ; Range: $[-1, +∞)$ ; Axis intercepts: ${(2, 0), (4, 0), (0, 8)}$ ; Asymptotes: $\emptyset$