Functions with variations and solving equations: 2nd grade course in PDF

Doing a course on functions in 10th grade is always essential for student progress. Thus, you will see in this course the definition, the direction of variation and the representative curve of a function as well as the study of its direction of variation.
The student should be able to study a function defined either by its algebraic expression or by its representative curve. He/she should also be able to represent the table of variation and solve equations and inequations graphically.

I. Reminders and complements on numerical functions

1. concept of function


We define a function on a set D\subset,\mathbb{R} by associating to each real x of D

a single real denoted by f(x).

We note: f:x,\mapsto  ,f(x)

  • x is the variable;
  • f(x) is the image of x by the function f .
  • If y = f(x), then x is the antecedent of y by f.
  • D is the set of definition of the function, i.e. the set of numbers that have an image by f.


Function defined in ℝ by its literal expression :


Function of two variables :

We call x and z the base and height of a triangle.

Its area is given by the formula: A(x,z)=\frac{xz}{2}.

Function taking its values in ℕ :

Each real number greater than 1 is assigned the number of divisors of its integer part.

2. Graphical representation of functions


The representative curve C_f of a function f (or graphic representation) is the set of points in the plane with coordinates ( x ; y ) such that :

  • the abscissa x is a value of the definition set D;
  • the y-intercept is the image of x by f. Therefore y = f(x).

In other words M(x,y)\in\,C_f if and only if x\in\,D and y = f(x).


f is the function defined on \mathbb{R} by f\,(x)=-x^2\,+\,2x .

Draw its representative curve C_f after having completed a table of values.

Curve of a square function

II. Graphical resolution of equations :

1. Equation of the type f(x) = k (with k a real number)


The solutions of the equation f(x) = k are the abscissae of the points of intersection of the curve C_f and the horizontal line of equation y=k .

Equation of the type f (x) = k

On this representative curve of the function f, the equation f(x) = k has for unique solution the number a.


Solving the equation f(x) = g(x), where f and g are two numerical functions, is the same as finding the coordinates of the intersection points of these two curves.

III-Notion of variations on an interval

Consider a function f represented on the interval [-4 ;6] .

Variations of a function


A function is said to be increasing over an interval when the images of any two numbers in that interval are always in the same order as the starting numbers. Graphically, the curve “rises”.



A function is said to be decreasing over an interval when the images of any two numbers in that interval are always in the opposite order to the starting numbers. Graphically, the curve “goes down”.


Table of variations :

This is a table where we summarize the variations of the function :

Table of variation of the function f

Vocabulary :

An extremum is any maximum or minimum of the function on its set of definition.


Draw a curve that can represent the function defined on [-2 ; 4] such that :

  • f reaches its maximum on [-2 ; 4] in 4;
  • The point (-2; 0) is a point on the curve of f;
  • An antecedent of 4 is 3 by f;
  • The image of 1 is -0.5 by f.
  • The minimum of f on [-2 ; 4] is -1, reached in 0;
  • f (4) = 5;
  • f is decreasing on [-2 ; 0] and increasing on [3,5 ; 4];
  • 3 has two antecedents by f which are x = 2 and x = 3.5.

Curve of a function

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