4 ESO Academics - UNIT 09 - FUNCTIONS

Unit 09 May
1. DEFINITION OF A FUNCTION.
A Function is a relation between two Variables such that for every value of the
first, there is only one corresponding value of the second. We say that the second
variable is a Function of the first variable. The first variable is the Independent
Variable (usually 𝒙𝒙), and the second variable is the Dependent Variable (usually 𝒚𝒚).
The independent variable and the dependent variable are real numbers.
Example 1:
You know the formula for the area of a circle is 𝐴𝐴 = 𝜋𝜋𝑟𝑟2
. This is a function as each
value of the independent variable 𝑟𝑟 gives you one value of the dependent variable 𝐴𝐴.
Example 2:
In the equation 𝑦𝑦 = 𝑥𝑥2
, 𝒚𝒚 is a function of 𝒙𝒙, since for each value of 𝑥𝑥, there is only
one value of 𝑦𝑦
Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.1

Unit 09 May
We normally write Functions as 𝒇𝒇(𝒙𝒙), and read this as “function 𝒇𝒇 of 𝒙𝒙”.
For example, the function 𝑦𝑦 = 𝑥𝑥2
− 5𝑥𝑥 + 2, is also written as 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2
− 5𝑥𝑥 + 2 (y
and f(x) are the same).
The Value of the Function 𝒇𝒇(𝒙𝒙) when 𝒙𝒙 = 𝒂𝒂 is 𝒇𝒇(𝒂𝒂).
If 𝑓𝑓(𝑥𝑥) = 𝑥𝑥2
− 5𝑥𝑥 + 2, then 𝑓𝑓(2) = 22
− 5 ∙ 2 + 2 = −4
A good way of presenting a function is by Graphical Representation. Graphs
give us a visual picture of the function. Normally, the values of the independent
variable (generally the x-values) are placed on the horizontal axis, while the values of
the dependent variable (generally the y-values) are placed on the vertical axis.
MATH VOCABULARY: Function, Independent Variable, Dependent Variable, Graph.

Unit 09 May
2. ELEMENTARY FUNCTIONS.
LINEAR FUNCTIONS.2.1.
A function that can be graphically represented in the Cartesian Coordinate
Plane by a straight line is called a Linear Function. The equation of a linear function is
𝒚𝒚 = 𝒎𝒎𝒎𝒎 + 𝒃𝒃
𝒎𝒎 is the Slope of the line and 𝒃𝒃 is the y-intercept. Remember that if 𝒎𝒎 > 𝟎𝟎 ,
the line is an Increasing Function, and if 𝒎𝒎 < 𝟎𝟎 , the line is a Decreasing Function.
If 𝒎𝒎 = 𝟎𝟎 , the equation of the function 𝒚𝒚 = 𝒃𝒃 .This type of linear functions are
called Constant Functions. Their graphs are horizontal lines.

Unit 09 May
If 𝒃𝒃 = 𝟎𝟎 , the equation of the function is 𝒚𝒚 = 𝒎𝒎𝒎𝒎, This type of linear functions
are called Proportional Functions. The variable “ ” is directly proportional to “𝒚𝒚 ”. The𝒙𝒙
constant ratio 𝒎𝒎 = 𝒚𝒚/𝒙𝒙 is called Proportionality Constant (or constant of
proportionality). Their graphs pass through the point (𝟎𝟎, 𝟎𝟎).
If 𝒎𝒎 = 𝟏𝟏, the proportionality function is 𝒚𝒚 = 𝒙𝒙 , and it is Called Identity
Function. This line is the Angle Bisector of the first and third quadrants.

Unit 09 May
PARABOLAS AND QUADRATIC FUNCTIONS.2.2.
A function whose graph is a Parabola is called a Quadratic Function. The
equation of a quadratic function is:
𝒚𝒚 = 𝒂𝒂𝒙𝒙𝟐𝟐
+ 𝒃𝒃𝒃𝒃 + 𝒄𝒄, 𝒂𝒂 ≠ 𝟎𝟎
A Parabola will have either an Absolute Minimum or an Absolute Maximum.
This point is called the Vertex of the parabola. There is a Line of Symmetry which will
divide the graph into two halves. This line is called the Axis of Symmetry of the
parabola.

Unit 09 May
If two Quadratic Functions have the same “𝒂𝒂”, the corresponding parabolas are
equal, but they are placed in different positions.
The parabola will open upward or downward. If 𝒂𝒂 > 𝟎𝟎 , the parabola opens
Upward. If 𝒂𝒂 < 𝟎𝟎 , the parabola opens Downward.
The greater is |𝒂𝒂| , the slimmer the parabola will be:

Unit 09 May
A Parabola 𝒚𝒚 = 𝒂𝒂𝒙𝒙𝟐𝟐
+ 𝒃𝒃𝒃𝒃 + 𝒄𝒄 can be represented from these points:
Axes Intercept Points.
• 𝒙𝒙 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊: An x-intercept is a point on the graph where 𝒚𝒚 = 𝟎𝟎. If
𝒚𝒚 = 𝟎𝟎 ⇒ 𝒂𝒂𝒙𝒙𝟐𝟐
+ 𝒃𝒃𝒃𝒃 + 𝒄𝒄 = 𝟎𝟎. When we solve the equation we can have:
 Two different real solutions: 𝒙𝒙𝟏𝟏; 𝒙𝒙𝟐𝟐. Then there are two x-intercept points
(𝒙𝒙𝟏𝟏, 𝟎𝟎) and (𝒙𝒙𝟐𝟐, 𝟎𝟎).
 One double real solution: 𝒙𝒙𝟏𝟏 = 𝒙𝒙𝟐𝟐. Then there is only one x-intercept
point: (𝒙𝒙𝟏𝟏, 𝟎𝟎).
 No real solutions. Then the graph does not intercept the x-axis.

Unit 09 May
 To summarize we can say that it will depends on the Discriminant:
• 𝒚𝒚 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊: is a point on the graph where 𝒙𝒙 = 𝟎𝟎. If 𝒙𝒙 = 𝟎𝟎 ⇒ 𝒚𝒚 = 𝒄𝒄. Then
the y-intercept point is (𝒄𝒄, 𝟎𝟎)

Unit 09 May
Vertex �𝑽𝑽𝒙𝒙, 𝑽𝑽𝒚𝒚�.
𝑽𝑽𝒙𝒙 =
−𝒃𝒃
𝟐𝟐𝟐𝟐
To find 𝑽𝑽𝒚𝒚 we need to calculate:
𝑽𝑽𝒚𝒚 = 𝒇𝒇(𝑽𝑽𝒙𝒙) = 𝒂𝒂𝑽𝑽𝒙𝒙
𝟐𝟐
+ 𝒃𝒃𝑽𝑽𝒙𝒙 + 𝒄𝒄
Once we have these
points we can Plot the graph:
The Basic Parabola is 𝒚𝒚 = 𝒙𝒙𝟐𝟐
. The function is symmetrical about the x-axis. Its
vertex is the point (𝟎𝟎, 𝟎𝟎) , which is also the absolute minimum. The graph has two
branches (one of them is decreasing and the other one is increasing).

Unit 09 May
INVERSELY PROPORTIONAL FUNCTIONS.2.3.
If the variables “𝒚𝒚” and “𝒙𝒙” are Inversely Proportional, then the functional
dependence between them is represented by the equation:
𝒚𝒚 =
𝒌𝒌
𝒙𝒙
; 𝒌𝒌 = 𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄𝒄 ≠ 𝟎𝟎
Let´s start from the easiest one:
𝒚𝒚 =
𝟏𝟏
𝒙𝒙
Its graph is a Hyperbola. It has two branches. If we focus on the branch for > 𝟎𝟎
: As 𝒙𝒙 increases, then 𝒚𝒚 decreases to 𝟎𝟎. As 𝒙𝒙 drops to 𝟎𝟎, then y increases to +∞. The
𝒙𝒙 and 𝒚𝒚 − 𝒂𝒂𝒂𝒂𝒂𝒂𝒂𝒂 are Asymptotes of the function. Asymptote is a line that a graph gets
closer and closer to, but never touches or crosses it.
In the general case the Inversely Proportional Functions are:
𝒚𝒚 =
𝒌𝒌
𝒙𝒙

Unit 09 May
They are Hyperbolas whose Asymptotes are the coordinate axes:
RATIONAL FUNCTIONS.2.4.
The Inversely Proportional Functions are a particular case of Rational
Functions. We will study the easiest case which equation is:
𝒚𝒚 =
𝒌𝒌
𝒙𝒙 − 𝒂𝒂
+ 𝒃𝒃
If 𝒂𝒂 = 𝒃𝒃 = 𝟎𝟎 ⟹ 𝑰𝑰. 𝑷𝑷. 𝑭𝑭. They are Hyperbolas whose Asymptotes AREN´T the
coordinate axes:
The Asymptotes will depends on the values of 𝒂𝒂 and 𝒃𝒃

Unit 09 May
To plot the graph we have to know 𝒂𝒂, 𝒃𝒃 and 𝒌𝒌. and find the Asymptotes and
draw them. Then we look for values on each branch with the help of a table. We study
the function taking in care of the value of 𝒂𝒂.

Unit 09 May
RADICAL FUNCTIONS.2.5.
A Radical Function is any function that contains a variable inside a Root. This
includes square roots, cubed roots, or any nth
root.
𝒚𝒚 = 𝒂𝒂√𝒃𝒃𝒃𝒃 + 𝒄𝒄
𝒏𝒏
Let´s start with the easiest one:
𝒚𝒚 = √𝒙𝒙
It is half a Parabola. If we square both sides of the function and isolate 𝒙𝒙, we
end up with the equation of the parabola in terms of 𝒚𝒚.
𝒚𝒚 = √𝒙𝒙 ⇒ 𝒚𝒚𝟐𝟐
= 𝒙𝒙
The functions 𝒚𝒚 = 𝒂𝒂√𝒙𝒙 + 𝒃𝒃 and 𝒚𝒚 = 𝒂𝒂√−𝒙𝒙 + 𝒃𝒃 are also half parabolas.

Unit 09 May
EXPONENTIAL FUNCTIONS.2.6.
Do you remember Compound Interest problems? This is an example of
Exponential Function (the variable “𝒕𝒕” is at the exponent of a power).
The easiest one is: 𝒚𝒚 = 𝒂𝒂𝒙𝒙
. The base “𝒂𝒂” can be any positive real number,
𝒂𝒂 ≠ 𝟏𝟏. Look at these graphs:
The graphs of the functions passes through the points (𝟎𝟎, 𝟏𝟏) and (𝟏𝟏, 𝒂𝒂). The
functions 𝒚𝒚 = 𝒚𝒚𝟎𝟎 𝒂𝒂𝒌𝒌𝒌𝒌
; 𝒚𝒚𝟎𝟎, 𝒌𝒌 ∈ ℝ , are also exponential functions. Their graphs are
similar to the graph of 𝒚𝒚 = 𝒂𝒂𝒙𝒙
.
The best thing about exponential functions is that they are so useful in real
world situations. Exponential Functions are used to model populations, carbon date
artifacts, help coroners determine time of death, compute investments, as well as
many other applications. If 𝒂𝒂 > 𝟏𝟏, the function is increasing and if 𝒂𝒂 < 𝟏𝟏 , the function
is decreasing.
The functions with the equation:

Unit 09 May
𝒚𝒚 = 𝒂𝒂𝒙𝒙
+ 𝒃𝒃
Are also Exponential Functions. The graph 𝒚𝒚 = 𝒂𝒂𝒙𝒙
+ 𝒃𝒃 can be obtained by
scrolling vertically the graph from the function 𝒚𝒚 = 𝒂𝒂𝒙𝒙
.
The functions with the equation:
𝒚𝒚 = 𝒂𝒂(𝒙𝒙+𝒃𝒃)
Are also Exponential Functions.
The graph 𝒚𝒚 = 𝒂𝒂(𝒙𝒙+𝒃𝒃)
can be obtained by moving horizontally the graph from
the function 𝒚𝒚 = 𝒂𝒂𝒙𝒙
.

Unit 09 May

Unit 09 May
LOGARITHMIC FUNCTIONS.2.7.
The functions 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂 𝒙𝒙 are called Logarithmic Functions. The base “ ” can𝒂𝒂
be any positive real number, 𝒂𝒂 ≠ 𝟏𝟏 .
Look at the graphs of 𝒚𝒚 = 𝟐𝟐𝒙𝒙
and 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝟐𝟐 𝒙𝒙:
In general, if we have two functions, 𝒇𝒇(𝒙𝒙) and 𝒈𝒈(𝒙𝒙 , where if) (𝒂𝒂, 𝒃𝒃) lies on the
graph of 𝒇𝒇(𝒙𝒙 , then the point) (𝒃𝒃, 𝒂𝒂) lies on the graph of 𝒈𝒈(𝒙𝒙 , we say that) is the𝒇𝒇
Inverse Function of and vice versa. The Inverse Function of𝒈𝒈 is denoted by𝒇𝒇 𝒇𝒇−𝟏𝟏
(read f inverse, not to be confused with exponentiation).
The graphs of the functions 𝒚𝒚 = 𝟐𝟐𝒙𝒙
and 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝟐𝟐 𝒙𝒙 are symmetric with respect
to the line 𝒚𝒚 = . In general, graphs of inverse functions,𝒙𝒙 and𝒇𝒇 𝒇𝒇−𝟏𝟏
are symmetric
with respect to the line 𝒚𝒚 = .𝒙𝒙
Look now at the graphs of 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝟐𝟐 𝒙𝒙 and 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝟏𝟏
𝟐𝟐
𝒙𝒙:

Unit 09 May
We see that the graph of 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂 𝒙𝒙 passes through the points (𝟏𝟏, 𝟎𝟎) and
(𝒂𝒂 , 𝟏𝟏) If. 𝒂𝒂 > 𝟏𝟏 the graph will be more closed than if is greater. If𝒂𝒂 𝟎𝟎 < 𝒂𝒂 < 𝟏𝟏 the
graph will be more closed than if is smaller. If𝒂𝒂 𝒂𝒂 > , the function is increasing and if𝟏𝟏
𝒂𝒂 < , the function is decreasing.𝟏𝟏
If we have the function 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂 𝒙𝒙 + 𝒃𝒃, we obtain the graph scrolling the
graph 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂 𝒙𝒙. If 𝒃𝒃 > 𝟎𝟎 the graph is scrolling up units, if𝒃𝒃 𝒃𝒃 < 𝟎𝟎 the graph is
scrolling down units.𝒃𝒃

Unit 09 May
If we have the function 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂(𝒙𝒙 + 𝒃𝒃) we obtain the graph obtained by
moving horizontally the graph 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂 𝒙𝒙. If 𝒃𝒃 > 𝟎𝟎 the graph is moving left units, if𝒃𝒃
𝒃𝒃 < 𝟎𝟎 the graph is moving right units.𝒃𝒃

Unit 09 May
TRIGONOMETRIC FUNCTIONS.2.8.
The Trigonometric Function 𝒚𝒚 = 𝒔𝒔𝒔𝒔𝒔𝒔 𝒙𝒙 give for any angle measured in radians,
its sine value.
The Trigonometric Function 𝒚𝒚 = 𝒄𝒄𝒄𝒄𝒄𝒄 𝒙𝒙 give for any angle measured in radians,
its cosine value.
MATH VOCABULARY: Cartesian Coordinate Plane, Linear Function, Slope, Increasing
Function, Decreasing Function, Constant Function, Proportional Function, Identity
Function, Angle Bisector, Parabola, Quadratic Function, Absolute Minimum, Absolute
Maximum, Vertex, Line of Symmetry, Axis of Symmetry, To Plot, Inversely Proportional
Function, Hyperbola, Asymptotes, Rational Function, Radical Function, Exponential
Function, Logarithmic Function, Inverse Function, Trigonometric Function.

Unit 09 May
3. DOMAIN AND RANGE.
The Domain of a function is the complete set of possible values of the
independent variable in the function. The Range (or Image) of a function is the
complete set of all possible resulting values of the dependent variable of a function,
after we have substituted the values in the domain.
𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫 𝒐𝒐𝒐𝒐 𝒇𝒇 = 𝑫𝑫𝑫𝑫𝑫𝑫𝑫𝑫
𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰 𝒐𝒐𝒐𝒐 𝒇𝒇 = 𝑰𝑰𝑰𝑰𝑰𝑰
LINEAR FUNCTIONS.3.1.
The Domain of a Linear Function is . The Range is usually alsoℝ ℝ Only if.
𝒚𝒚 = 𝒃𝒃 the Range is, [𝒃𝒃, 𝒃𝒃]

Unit 09 May
PARABOLAS AND QUADRATIC FUNCTIONS.3.2.
The Domain of a Quadratic Function is . The Range is depending of theℝ
Vertex position.
INVERSELY PROPORTIONAL FUNCTIONS.3.3.
The Domain of a Inversely Proportional Function is ℝ − {𝟎𝟎 . The Range is also}
ℝ − {𝟎𝟎}.

Unit 09 May
RATIONAL FUNCTIONS.3.4.
Remember that
𝒚𝒚 =
𝒌𝒌
𝒙𝒙 − 𝒂𝒂
+ 𝒃𝒃
The Domain is ℝ − {𝒂𝒂 . The Range is} ℝ − {𝒃𝒃}.

Unit 09 May
RADICAL FUNCTIONS.3.5.
Remember that:
𝒚𝒚 = 𝒂𝒂√𝒃𝒃𝒃𝒃 + 𝒄𝒄
𝒏𝒏
The Domain of Radical Functions depends on the value of on the radicand.𝒄𝒄
The Range in the functions seen is always [𝟎𝟎, ∞) if 𝒂𝒂 > 𝟎𝟎 and (−∞, 𝟎𝟎] if 𝒂𝒂 < 𝟎𝟎.
EXPONENTIAL FUNCTIONS.3.6.
The Domain of the functions 𝒚𝒚 = 𝒂𝒂𝒙𝒙
, 𝒚𝒚 = 𝒂𝒂𝒙𝒙
+ 𝒃𝒃 and 𝒚𝒚 = 𝒂𝒂(𝒙𝒙+𝒃𝒃)
is ℝ The.
the functions 𝒚𝒚 = 𝒂𝒂𝒙𝒙
and 𝒚𝒚 = 𝒂𝒂(𝒙𝒙+𝒃𝒃)
isRange of [𝟎𝟎, ∞), the of 𝒚𝒚 = 𝒂𝒂𝒙𝒙
+ 𝒃𝒃 willRange
depends on the value of .𝒃𝒃

Unit 09 May
LOGARITHMIC FUNCTIONS.3.7.
If we have the function 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂(𝒙𝒙 + 𝒃𝒃) the Domain is depending on the
value of .𝒃𝒃 The function 𝒚𝒚 = 𝒍𝒍𝒍𝒍 𝒍𝒍𝒂𝒂 𝒙𝒙 + 𝒃𝒃 has as Domain (𝟎𝟎, ∞). The Range of all of
them will be ℝ.
The domain of is the range of𝒇𝒇 𝒇𝒇−
, and vice versa, the range of𝟏𝟏
is the𝒇𝒇
domain of 𝒇𝒇−𝟏𝟏
.

Unit 09 May
TRIGONOMETRIC FUNCTIONS.3.8.
The Domain of 𝒚𝒚 = 𝒔𝒔𝒔𝒔𝒔𝒔 𝒙𝒙 and 𝒚𝒚 = 𝒄𝒄𝒄𝒄𝒄𝒄 𝒙𝒙 is ℝ The Range is always [−𝟏𝟏, 𝟏𝟏]..
MATH VOCABULARY: Domain, Range, Image.
4. CONTINUOUS AND DISCONTINUOUS FUNCTIONS.
Consider the graph of 𝒚𝒚 = 𝒄𝒄𝒄𝒄𝒄𝒄 𝒙𝒙:
We can see that there are no “gaps” in the curve. Any value of “ ” will give us a𝒙𝒙
corresponding value of “ ”. We could continue the graph in the negative and positivey
directions, and we would never need to take the pencil off the paper. Such functions
are called Continuous Functions.

Unit 09 May
Now consider the function
𝒚𝒚 =
𝒙𝒙
𝒙𝒙 − 𝟐𝟐
We can see that the curve is discontinuous at 𝒙𝒙 = 𝟐𝟐 We observe that a small.
change in 𝒙𝒙 near to 𝒙𝒙 = , gives a very large change in the value of the function.𝟐𝟐
x y
1.99 -199
2.01 201
For a function to be Continuous at a point, the function must exist at the point
and any small change in “ ” produces only a small change in “𝒙𝒙 𝒇𝒇(𝒙𝒙 ”. If a function is not)
continuous at a point, we say that it is Discontinuous at that point.
A function 𝒇𝒇 is Continuous on the Open Interval (𝒂𝒂, 𝒃𝒃 if) is continuous at𝒇𝒇
every point in (𝒂𝒂, 𝒃𝒃 . There are different reasons why a function is Discontinuous at a)
point. The four functions below are discontinuous at 𝒙𝒙 = 𝟐𝟐.

Unit 09 May
The function has a “Finite Jump”.
The function is “Missing” a point.
The function has an “Infinite Jump”.
The function has a “Moved” point.

Unit 09 May
All the functions seen are Continuous except the Rational Functions that are
Discontinuous. The discontinuous point will be in the asymptote point.
MATH VOCABULARY: Continuous Function, Discontinuous Function, Finite Jump,
Infinite Jump.
5. INTERSECTION POINTS WITH THE AXIS.
The Intersection Points, are the 𝒙𝒙 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 and 𝒚𝒚 − points.𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊
The 𝒙𝒙 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 points can be calculated by solving the equation when 𝒚𝒚 = 𝟎𝟎 And.
the 𝒚𝒚 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 points can be calculated by solving the equation when 𝒙𝒙 = 𝟎𝟎.
The 𝒙𝒙 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 points are always (𝒂𝒂, 𝟎𝟎 , and the) 𝒚𝒚 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 are
always (𝟎𝟎, 𝒃𝒃).
Example 1:
𝑦𝑦 = −5𝑥𝑥 + 2
𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → 𝑓𝑓(𝑥𝑥) = 0
0 = −5𝑥𝑥 + 2
𝑥𝑥 =
2
5
𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → �
2
5
, 0�
𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → 𝑓𝑓(0)
𝑦𝑦 = −5 ∙ 0 + 2 = 2
𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → (0,2)
Example 2:
𝑦𝑦 = 𝑥𝑥2
+ 𝑥𝑥 − 6

Unit 09 May
𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → 𝑓𝑓(𝑥𝑥) = 0
0 = 𝑥𝑥2
+ 𝑥𝑥 − 6
𝑥𝑥 =
−1 ± �12 − 4 ∙ 1 ∙ (−6)
2
= �
𝑥𝑥1 = 2
𝑥𝑥2 = −3
𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → (2,0) 𝑎𝑎𝑎𝑎𝑎𝑎 (−3,0)
𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 → 𝑓𝑓(0)
𝑦𝑦 = 02
+ 0 − 6 = −6
𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → (0,6)
Remember that in Exponential Functions like 𝒇𝒇(𝒙𝒙) = 𝒂𝒂 , the𝒙𝒙
𝒚𝒚 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is
always (𝟎𝟎, 𝟏𝟏 and in Logarithmic Functions as) 𝒇𝒇(𝒙𝒙) = 𝐥𝐥𝐥𝐥𝐥𝐥 𝒂𝒂 𝒙𝒙 the 𝒙𝒙 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is
always (𝟏𝟏, 𝟎𝟎) In the Basic Sine Function the. 𝒙𝒙 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 points are those whose
𝐬𝐬𝐬𝐬 𝐬𝐬 𝜽𝜽 = , and the𝟎𝟎 𝒚𝒚 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is always (𝟎𝟎, 𝟎𝟎) In the Basic Cosine Function the.
𝒙𝒙 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 points are those whose 𝐜𝐜𝐜𝐜𝐜𝐜 𝜽𝜽 = , and the𝟎𝟎 𝒚𝒚 − 𝒊𝒊 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is always
(𝟎𝟎, 𝟏𝟏).
MATH VOCABULARY: Intersection Points.
6. VARIATIONS IN A FUNCTION.
INCREASING AND DECREASING.6.1.
A function is Increasing on an interval𝒇𝒇 (𝒂𝒂, 𝒃𝒃) if for any 𝒙𝒙𝟏𝟏 and 𝒙𝒙 in the𝟐𝟐
interval such that 𝒙𝒙𝟏𝟏 < 𝒙𝒙𝟐𝟐 then 𝒇𝒇(𝒙𝒙𝟏𝟏) < 𝒇𝒇(𝒙𝒙𝟐𝟐 . Another way to look at this is: as you)
trace the graph from to𝒂𝒂 (that is from left to right) the graph should go up.𝒃𝒃

Unit 09 May
A function is Decreasing on an interval𝒇𝒇 (𝒂𝒂, 𝒃𝒃) if for any 𝒙𝒙𝟏𝟏 and 𝒙𝒙 in the𝟐𝟐
interval such that 𝒙𝒙𝟏𝟏 < 𝒙𝒙𝟐𝟐 then 𝒇𝒇(𝒙𝒙𝟏𝟏) > 𝒇𝒇(𝒙𝒙𝟐𝟐 . Another way to look at this is: as you)
trace the graph from to𝒂𝒂 (that is from left to right) the graph should go down.𝒃𝒃

Unit 09 May

Unit 09 May
MAXIMA AND MINIMA.6.2.
A function has a Relative (or Local) Maximum at a point if its ordinate is𝒇𝒇
greater that the ordinates of the points around it. A function has a Relative (or Local)𝒇𝒇
Minimum at a point if its ordinate is smaller than the ordinates of the points around it.
A function has an Absolute (or Global) Maximum at a point if its ordinate is𝒇𝒇
the largest value that the function takes on the domain that we are working on. A
function has an Absolute (or Global) Minimum at a point if its ordinate is smallest𝒇𝒇
value that the function takes on the domain that we are working on.
MATH VOCABULARY: Increasing Function Decreasing Function, Relative Maximum,
Relative Maximum, Absolute Maximum, Absolute Minimum.

Unit 09 May
7. PERIODIC FUNCTIONS.
A Periodic Function repeats Cycle may begin at any point on the graph of the
function. The Period of a function is the horizontal length a pattern of 𝒚𝒚 − 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗 at
regular intervals. One complete pattern is a Cycle.
If 𝒇𝒇 is a Periodic Function whose Period is 𝑷𝑷, then 𝒇𝒇(𝒙𝒙 + 𝒌𝒌 ∙ 𝑷𝑷) = 𝒇𝒇(𝒙𝒙) for all
values of 𝒙𝒙.
The Amplitude of a periodic function measures the amount of variation in the
function values.
The Amplitude of a periodic function is half the difference between the
maximum and minimum values of the function.

Unit 09 May
The only Periodic Functions studied are the Trigonometric Functions seen.
MATH VOCABULARY: Periodic Function, Cycle, Period, Amplitude.

Unit 09 May
8. SYMMETRIC FUNCTIONS.
There are two kinds of Symmetric Functions:
• Symmetric Function respect to the Y-Axis: 𝒇𝒇(−𝒙𝒙) = 𝒇𝒇(𝒙𝒙). It is also called Even
Function.

Unit 09 May
• Symmetric Function respect to Origin: 𝒇𝒇(−𝒙𝒙) = −𝒇𝒇(𝒙𝒙). It is also called Odd
Function.
To study the symmetry of a function we have to calculate 𝒇𝒇(−𝒙𝒙) and compare
the result with 𝒇𝒇(𝒙𝒙).
MATH VOCABULARY: Symmetric Function, Even Function, Odd Function.

4 ESO Academics - UNIT 09 - FUNCTIONS

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