LLG
Advanced Math and Science Pilot Class
Mathematics, Grade 10
2014 – 2015
Paris – Abu Dhabi
Chapter 12 : Usual Functions
Let 𝑓 and 𝑔 be two functions defined on an interval 𝐼. We consider an orthogonal frame of the plane and 𝒞𝑓 and 𝒞𝑔
the graphs of 𝑓 and 𝑔 in that frame.
I-Variations of a function :
Definition :
 𝑓 is increasing on the interval 𝐼 IFF for all real numbers 𝑎, 𝑏 belonging to 𝐼, we have :
𝑎 < 𝑏 ⟺ 𝑓(𝑎) < 𝑓(𝑏)
Graphically speaking, the curve is climbing up over 𝐼.
 𝑓 is decreasing on the interval 𝐼 IFF for all real numbers 𝑎, 𝑏 belonging to 𝐼, we have :
𝑎 < 𝑏 ⟺ 𝑓(𝑎) > 𝑓(𝑏)
Graphically speaking, the curve is climbing down over 𝐼.
Table of variations :
We summarize the variations of a function in a table of variations as in the example below :
f reaches its maximum
on [-4 ; 6] at -2
On [4 ; 6] , f is increasing
On [-4 ;-2], f is increasing
f has a minimum over [-4 ; 6]
reached at 4
On [-2 ;4], f is decreasing
𝑥
−4
−2
4
1,3
6
-1,9
𝑓 (𝑥)
-0,9
The turning points of the function are
−2 and 4 : it’s where the function’s
variations change…
-4
We say that the function 𝑓 has a maximum in 𝑎 ∈ 𝐷𝑓 when for all 𝑥 ∈ 𝐷𝑓 we have 𝑓(𝑥) ≤ 𝑓(𝑎). In that case the
maximum is 𝑓(𝑎).
We say that the function 𝑓 has a minimum when 𝑥 = 𝑎 ∈ 𝐷𝑓 if for all 𝑥 ∈ 𝐷𝑓 we have 𝑓 (𝑥) ≥ 𝑓(𝑎). In that case
the minimum is 𝑓(𝑎).
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Grade 10
Chapter 12 : Usual Functions
Remark : To prove that 𝑀 ∈ ℝ is the maximum (resp. the minimum) of a function 𝑓 it is not sufficient to prove
that for all 𝑥 ∈ 𝐷𝑓 we have 𝑓(𝑥) ≤ 𝑀 (resp. 𝑓(𝑥) ≥ 𝑀), you also have to prove that there is 𝑎 ∈ 𝐷𝑓 such that
𝑓(𝑎) = 𝑀.
II-Linear functions :
Definition :
 For a function 𝑓 and two real numbers 𝑎 and 𝑏 (𝑎 ≠ 𝑏) in the domain, we call rate of change of 𝑓between 𝑎
and 𝑏 the number :
𝑓(𝑏)−𝑓(𝑎)
.
𝑏−𝑎
 Linear functions are functions defined on ℝ for which the rate of change is a constant (not depending on 𝑥).
Remark : If a function is increasing on an interval I then its rates of change on this interval are positive, if it is
decreasing then its rates of change are negative.
Properties :
A linear function has an algebraic expression of the form 𝑚𝑥 + 𝑏, with 𝑚, 𝑏 ∈ ℝ two constants.
Proof :
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The graph of a linear function is a straight line, wit equation 𝑦 = 𝑚𝑥 + 𝑏, with 𝑚, 𝑏 ∈ ℝ two
constants.
A linear function 𝑓(𝑥) = 𝑚𝑥 + 𝑏, is increasing if 𝑚 > 0.
A linear function 𝑓(𝑥) = 𝑚𝑥 + 𝑏, is decreasing if 𝑚 < 0.
𝑚>0
𝑥


p
m
𝑚<0

𝑥


p
m

𝑓(𝑥)
𝑓(𝑥)
0
0
[
[
A
A
t
t
Proof :
t
t
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i
i
r
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r
e
e
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z
z
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l
l
’
’
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a
a
t
2
t
t
Grade 10
Chapter 12 : Usual Functions
t
e
Examples : Find the expression of the linear function such that 𝑓(−3) = 1 and 𝑓(4) = −1 and draw its table of
variations and table of sign.
III-Square function and quadratic functions :
3.1. Square Function :
The square function is defined over ℝ, by 𝑓(𝑥) = 𝑥 2 .
 Its curve is called a parabola, its vertex is (0, 0).
 The square function is decreasing on ]−∞ ; 𝟎] and increasing on [𝟎 ; +∞[.
 The square function is an even function 𝑖𝑒 𝑓(−𝑥) = 𝑓(𝑥), for all 𝑥 ∈ ℝ .
Proof :
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Consequences :
 𝑥 < 𝑦 is not always equivalent to 𝑥 2 < 𝑦 2 !!!!!
 The parabola is symmetrical about the 𝑦 − 𝑎𝑥𝑖𝑠.
𝑥
−∞
0
+∞
𝑥
𝑓(𝑥)
𝑓(𝑥) = 𝑥 2
0
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Grade 10
Chapter 12 : Usual Functions
3.2. Quadratic Function :
A quadratic function is defined on ℝ, by 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 where 𝑎, 𝑏, 𝑐 are constant, real numbers.
Its curve is called a parabola, it has a vertex, and opens either upwards or downwards.
If 𝑎 > 0 the parabola opens upwards and
the vertex will be a minimum point.
𝑥
−∞
−
𝑏
2𝑎
If 𝑎 < 0 the parabola opens downwards and
the vertex will be a maximum point.
𝑥
+∞
−∞
−
𝑏
2𝑎
𝑓 (−
𝑓(𝑥)
𝑓 (−
𝑏
)
2𝑎
𝑓(𝑥)
+∞
𝑏
)
2𝑎
If you imagine a vertical line running through the vertex, you will notice the graph is symmetrical on the left and right
sides of this line. The axis is called an axis of symmetry of the parabola.
A quadratic function can be written on its “turning-point” form, or standard form : 𝑓(𝑥) = 𝑎(𝑥 − 𝛼)2 + 𝛽 .
 In that case the vertex is (𝜶, 𝜷), and you have to know how to prove its variations.
Rmk : If the standard form is not given , you can use your GDC to work it out.
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Example : Give the variations of the functions defined by 𝑓(𝑥) = −2(𝑥 + 1)2 + 5 and 𝑔(𝑥) = 2 (𝑥 − 1)2 − 4, find
the vertex and the axis of symmetry of their curves and sketch the curves, labelling the vertex and the 𝑦 −intercept.
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Grade 10
Chapter 12 : Usual Functions
IV. Reciprocal Function :
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The reciprocal function is defined over ℝ∗, by 𝑓(𝑥) = 𝑥.
 Its curve is called a hyperbola with center is (0, 0).
 The reciprocal function is decreasing on ]−∞ ; 𝟎] and also decreasing on [𝟎 ; +∞[.
 The reciprocal function is an odd function 𝑖𝑒 𝑓(−𝑥) = 𝑓(𝑥), for all 𝑥 ∈ ℝ∗ .
Proof :
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Consequences :

1
𝑥
1
< 𝑦 is not always equivalent to 𝑥 > 𝑦 !!!!! It’s true IFF ………………………………………………………………………….......
 The reciprocal function is not decreasing on ℝ∗ !!!!
 The hyperbola is symmetrical about the origin of the frame.
−∞
𝑥
𝑓(𝑥) =
1
𝑥
0
+∞
𝑥
1
4
1
3
1
2
1
2
3
4
𝑓(𝑥)
5
Grade 10
Chapter 12 : Usual Functions
APPLICATION :
4
(a) Let’s work out the domain and the variations of the function 𝑓 defined by 𝑓(𝑥) = 3 − 𝑥−1.
(b) Same questions with 𝑔(𝑥) =
3
𝑥+2
− 5.
V. Square root Function :
The square root function is defined on ℝ+ , by 𝑓(𝑥) = √𝑥.
 Its curve is a half parabola.
 The square root function is increasing on ℝ+ .
 The reciprocal function is neither odd nor even, as its domain is not symmetrical with respect to 0.
Proof :
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0
+∞
0
𝑓(𝑥) = √𝑥
0
VI. Cube Function :
The cube function is defined on ℝ, by 𝑓(𝑥) = 𝑥 3 .
 Its curve is called a cubic.
 The cube function is increasing on ℝ.
 The reciprocal function is odd, 𝑓(−𝑥) = −𝑓(𝑥).
Proof : on the notebook.
0
0
+∞
𝑓(𝑥) = √𝑥
0
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Grade 10
Chapter 12 : Usual Functions