study f(x)=x^2-3x

We save the definition:

f (x) = x^{2} - 3 x

Let's examine the function

f (x) = x^{2} - 3 x

1) Let's find the domain of

x^{2} - 3 x

ⓘ This expression imposes no restriction on the real variable

Domain of definition:

x \in ℝ

2) Let's examine the parity of

x^{2} - 3 x

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

: f(-

x

) =

{(- x)}^{2} - 3 \cdot (- x)

Let's evaluate:

{(- x)}^{2} - 3 \cdot (- x)

ⓘ Calculate powers before multiplications, additions and subtractions. If the integer exponent is even, the minus sign of the base disappears; if it is odd, it goes in front of the power:

{(- x)}^{2}

x^{2}

■A pair of minus signs cancels out:

-3 \cdot (- x)

3 x

x^{2} + 3 x

f(-

x

) - f(

x

) =

6 x

and f(-

x

) + f(

x

) =

2 x^{2}

are nonzero polynomials: the function is neither even nor odd

Parity: neither even nor odd

3) Differentiability

⇥ⓘ Differentiate a sum by differentiating each term, then adding the derivatives:

\frac{d}{d x} (x^{2} - 3 x)

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 3 x)

⇥

\frac{d}{d x} (x^{2}) + \frac{d}{d x} (- 3 x)

ⓘ The derivative of

x^{n}

n x^{n - 1}

\frac{d}{d x} (x^{2})

2 x

■A constant factor stays in front of the derivative: that of

k u

k \cdot u'

\frac{d}{d x} (- 3 x)

-3

Derivative:

2 x - 3

4) Let's examine the variations of

f (x)

: first find where its derivative

f^{'} (x) = 2 x - 3

is zero

We solve in

ℝ

2 x - 3 = 0

ⓘ We add

3

to both sides

2 x = 3

ⓘ Divide both sides by

2

Solution found:

x = \frac{3}{2}

f^{'} (x)

is zero at

x = \frac{3}{2}

Built from usual functions,

f^{'} (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

f^{'} (0) = -3

< 0:

f

is decreasing on

(-∞, \frac{3}{2})

f^{'} (2) = 1

> 0:

f

is increasing on

(\frac{3}{2}, +∞)

Let's calculate the values of

f

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

x

tends to

-∞

f (x)

tends to

+∞

f (\frac{3}{2}) = {(\frac{3}{2})}^{2} - 3 \cdot \frac{3}{2} = - \frac{9}{4}

x

tends to

+∞

f (x)

tends to

+∞

Let's construct the variation table:

$x$	$-∞$		$\frac{3}{2}$		$+∞$
$f^{'} (x)$		-	$0$	+
$f (x)$	$+∞$	↘	$- \frac{9}{4}$	↗	$+∞$

f

is decreasing on

(-∞, \frac{3}{2}]

and increasing on

[\frac{3}{2}, +∞)

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

⇥On

ℝ

f

is continuous: it takes every value of

[- \frac{9}{4}, +∞)

Range:

[- \frac{9}{4}, +∞)

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

x^{2} - 3 x = 0

We solve in

ℝ

x^{2} - 3 x = 0

You can factorize the expression:

x^{2} - 3 x

into:

x \cdot (x - 3)

A product is zero if and only if one of its factors is zero: we solve each factor equal to 0

⇥6.1) We solve in

ℝ

x = 0

⇥Solution found:

x = 0

⇥6.2) We solve in

ℝ

x - 3 = 0

⇥Solution found:

x = 3

On the vertical axis,

x = 0

: calculate f(0)

⇥

0^{2} - 3 \cdot 0

⇥ⓘ If a factor is zero, the product is zero:

-3 \cdot 0

0

■Adding 0 preserves the original value:

0 + 0

0

⇥

0

Axis intercepts:

{(0, 0), (3, 0)}

7) Let's determine the asymptotes of

x^{2} - 3 x

, for real

x

x^{2} - 3 x

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+-+3*x&v=x&b=-10.0&e=10.0&min=-10.175&max=96.925

Asymptotes:

\emptyset

▶Study of $f (x)$ : Domain of definition: $x \in ℝ$ ; Parity: neither even nor odd; Derivative: $2 x - 3$ ; Range: $[- \frac{9}{4}, +∞)$ ; Axis intercepts: ${(0, 0), (3, 0)}$ ; Asymptotes: $\emptyset$

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