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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.2

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 = 𝒎𝒎𝒙𝒙 + 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.

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Unit 09 May

If 𝒃𝒃 = 𝟎𝟎 , the equation of the function is 𝒚𝒚 = 𝒎𝒎𝒙𝒙, This type of linear functions

are called Proportional Functions. The variable “ ” is directly proportional to “𝒚𝒚 ”. The 𝒙𝒙

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.4

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 + 𝒃𝒃𝒙𝒙 + 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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.5

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:

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.6

Unit 09 May

A Parabola 𝒚𝒚 = 𝒂𝒂𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒙𝒙 + 𝒄𝒄 can be represented from these points:

Axes Intercept Points.

• 𝒙𝒙 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊: An x-intercept is a point on the graph where 𝒚𝒚 = 𝟎𝟎. If

𝒚𝒚 = 𝟎𝟎 ⇒ 𝒂𝒂𝒙𝒙𝟐𝟐 + 𝒃𝒃𝒙𝒙 + 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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.7

Unit 09 May

To summarize we can say that it will depends on the Discriminant:

• 𝒚𝒚 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊: is a point on the graph where 𝒙𝒙 = 𝟎𝟎. If 𝒙𝒙 = 𝟎𝟎 ⇒ 𝒚𝒚 = 𝒄𝒄. Then

the y-intercept point is (𝒄𝒄,𝟎𝟎)

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.8

Unit 09 May

Vertex �𝑽𝑽𝒙𝒙,𝑽𝑽𝒚𝒚�.

𝑽𝑽𝒙𝒙 =−𝒃𝒃𝟐𝟐𝒂𝒂

To find 𝑽𝑽𝒚𝒚 we need to calculate:

𝑽𝑽𝒚𝒚 = 𝒇𝒇(𝑽𝑽𝒙𝒙) = 𝒂𝒂𝑽𝑽𝒙𝒙𝟐𝟐 + 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).

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.9

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 =𝒌𝒌𝒙𝒙

; 𝒌𝒌 = 𝒄𝒄𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊𝒂𝒂𝒊𝒊𝒊𝒊 ≠ 𝟎𝟎

Let´s start from the easiest one:

𝒚𝒚 =𝟏𝟏𝒙𝒙

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 − 𝒂𝒂𝒙𝒙𝒊𝒊𝒊𝒊 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 =𝒌𝒌𝒙𝒙

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.10

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

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.11

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.12

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√𝒃𝒃𝒙𝒙 + 𝒄𝒄𝒊𝒊

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.13

Unit 09 May

EXPONENTIAL FUNCTIONS. 2.6.

Do you remember Compound Interest problems? This is an example of

Exponential Function (the variable “𝒊𝒊” 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 = 𝒚𝒚𝟎𝟎𝒂𝒂𝒌𝒌𝒙𝒙; 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:

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.14

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.15

Unit 09 May

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.16

Unit 09 May

LOGARITHMIC FUNCTIONS. 2.7.

The functions 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂 𝒙𝒙 are called Logarithmic Functions. The base “ ” can 𝒂𝒂

be any positive real number, 𝒂𝒂 ≠ 𝟏𝟏 .

Look at the graphs of 𝒚𝒚 = 𝟐𝟐𝒙𝒙 and 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝟐𝟐 𝒙𝒙:

In general, if we have two functions, 𝒇𝒇(𝒙𝒙) and 𝒍𝒍(𝒙𝒙 , where if ) (𝒂𝒂,𝒃𝒃) lies on the

graph of 𝒇𝒇(𝒙𝒙 , then the point ) (𝒃𝒃,𝒂𝒂) lies on the graph of 𝒍𝒍(𝒙𝒙 , we say that ) is the 𝒇𝒇

Inverse Function of and vice versa. The Inverse Function of 𝒍𝒍 is denoted by 𝒇𝒇 𝒇𝒇−𝟏𝟏

(read f inverse, not to be confused with exponentiation).

The graphs of the functions 𝒚𝒚 = 𝟐𝟐𝒙𝒙 and 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝟐𝟐 𝒙𝒙 are symmetric with respect

to the line 𝒚𝒚 = . In general, graphs of inverse functions, 𝒙𝒙 and 𝒇𝒇 𝒇𝒇−𝟏𝟏 are symmetric

with respect to the line 𝒚𝒚 = . 𝒙𝒙

Look now at the graphs of 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝟐𝟐 𝒙𝒙 and 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝟏𝟏𝟐𝟐𝒙𝒙:

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.17

Unit 09 May

We see that the graph of 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂 𝒙𝒙 passes through the points (𝟏𝟏,𝟎𝟎) and

(𝒂𝒂 ,𝟏𝟏) If . 𝒂𝒂 > 𝟏𝟏 the graph will be more closed than if is greater. If 𝒂𝒂 𝟎𝟎 < 𝒂𝒂 < 𝟏𝟏 the

graph will be more closed than if is smaller. If 𝒂𝒂 𝒂𝒂 > , the function is increasing and if 𝟏𝟏

𝒂𝒂 < , the function is decreasing. 𝟏𝟏

If we have the function 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂 𝒙𝒙 + 𝒃𝒃, we obtain the graph scrolling the

graph 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂 𝒙𝒙. If 𝒃𝒃 > 𝟎𝟎 the graph is scrolling up units, if 𝒃𝒃 𝒃𝒃 < 𝟎𝟎 the graph is

scrolling down units. 𝒃𝒃

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.18

Unit 09 May

If we have the function 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂(𝒙𝒙 + 𝒃𝒃) we obtain the graph obtained by

moving horizontally the graph 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂 𝒙𝒙. If 𝒃𝒃 > 𝟎𝟎 the graph is moving left units, if 𝒃𝒃

𝒃𝒃 < 𝟎𝟎 the graph is moving right units. 𝒃𝒃

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.19

Unit 09 May

TRIGONOMETRIC FUNCTIONS. 2.8.

The Trigonometric Function 𝒚𝒚 = 𝒊𝒊𝒊𝒊𝒊𝒊 𝒙𝒙 give for any angle measured in radians,

its sine value.

The Trigonometric Function 𝒚𝒚 = 𝒄𝒄𝒄𝒄𝒊𝒊 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. Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.20

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.

𝑫𝑫𝒄𝒄𝒎𝒎𝒂𝒂𝒊𝒊𝒊𝒊 𝒄𝒄𝒇𝒇 𝒇𝒇 = 𝑫𝑫𝒄𝒄𝒎𝒎𝒇𝒇

𝑰𝑰𝒎𝒎𝒂𝒂𝒍𝒍𝒊𝒊 𝒄𝒄𝒇𝒇 𝒇𝒇 = 𝑰𝑰𝒎𝒎𝒇𝒇

LINEAR FUNCTIONS. 3.1.

The Domain of a Linear Function is . The Range is usually also ℝ ℝ Only if .

𝒚𝒚 = 𝒃𝒃 the Range is, [𝒃𝒃,𝒃𝒃]

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.21

Unit 09 May

PARABOLAS AND QUADRATIC FUNCTIONS. 3.2.

The Domain of a Quadratic Function is . The Range is depending of the ℝ

Vertex position.

INVERSELY PROPORTIONAL FUNCTIONS. 3.3.

The Domain of a Inversely Proportional Function is ℝ− {𝟎𝟎 . The Range is also }

ℝ− {𝟎𝟎}.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.22

Unit 09 May

RATIONAL FUNCTIONS. 3.4.

Remember that

𝒚𝒚 =𝒌𝒌

𝒙𝒙 − 𝒂𝒂+ 𝒃𝒃

The Domain is ℝ − {𝒂𝒂 . The Range is } ℝ − {𝒃𝒃}.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.23

Unit 09 May

RADICAL FUNCTIONS. 3.5.

Remember that:

𝒚𝒚 = 𝒂𝒂√𝒃𝒃𝒙𝒙 + 𝒄𝒄𝒊𝒊

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 will Range

depends on the value of . 𝒃𝒃

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.24

Unit 09 May

LOGARITHMIC FUNCTIONS. 3.7.

If we have the function 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂(𝒙𝒙 + 𝒃𝒃) the Domain is depending on the

value of .𝒃𝒃 The function 𝒚𝒚 = 𝒍𝒍𝒄𝒄𝒍𝒍𝒂𝒂 𝒙𝒙 + 𝒃𝒃 has as Domain (𝟎𝟎,∞). The Range of all of

them will be ℝ .

The domain of is the range of 𝒇𝒇 𝒇𝒇− , and vice versa, the range of 𝟏𝟏 is the 𝒇𝒇

domain of 𝒇𝒇−𝟏𝟏 .

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.25

Unit 09 May

TRIGONOMETRIC FUNCTIONS. 3.8.

The Domain of 𝒚𝒚 = 𝒊𝒊𝒊𝒊𝒊𝒊 𝒙𝒙 and 𝒚𝒚 = 𝒄𝒄𝒄𝒄𝒊𝒊 𝒙𝒙 is ℝ The Range is always [−𝟏𝟏,𝟏𝟏]. .

MATH VOCABULARY: Domain, Range, Image.

4. CONTINUOUS AND DISCONTINUOUS FUNCTIONS.

Consider the graph of 𝒚𝒚 = 𝒄𝒄𝒄𝒄𝒊𝒊 𝒙𝒙:

We can see that there are no “gaps” in the curve. Any value of “ ” will give us a 𝒙𝒙

corresponding value of “ ”. We could continue the graph in the negative and positive y

directions, and we would never need to take the pencil off the paper. Such functions

are called Continuous Functions. Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.26

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 at 𝒇𝒇

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.27

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.28

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 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 and 𝒚𝒚 − points. 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊

The 𝒙𝒙 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 points can be calculated by solving the equation when 𝒚𝒚 = 𝟎𝟎 And .

the 𝒚𝒚 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 points can be calculated by solving the equation when 𝒙𝒙 = 𝟎𝟎.

The 𝒙𝒙 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 points are always (𝒂𝒂,𝟎𝟎 , and the) 𝒚𝒚 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 are

always (𝟎𝟎,𝒃𝒃).

Example 1:

𝑦𝑦 = −5𝑥𝑥 + 2

𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → 𝑓𝑓(𝑥𝑥) = 0

0 = −5𝑥𝑥 + 2

𝑥𝑥 =25

𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → �25

, 0�

𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → 𝑓𝑓(0)

𝑦𝑦 = −5 ∙ 0 + 2 = 2

𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → (0,2)

Example 2:

𝑦𝑦 = 𝑥𝑥2 + 𝑥𝑥 − 6

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.29

Unit 09 May

𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → 𝑓𝑓(𝑥𝑥) = 0

0 = 𝑥𝑥2 + 𝑥𝑥 − 6

𝑥𝑥 =−1 ± �12 − 4 ∙ 1 ∙ (−6)

2= � 𝑥𝑥1 = 2

𝑥𝑥2 = −3

𝑥𝑥 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → (2,0) 𝑎𝑎𝑖𝑖𝑎𝑎 (−3,0)

𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → 𝑓𝑓(0)

𝑦𝑦 = 02 + 0 − 6 = −6

𝑦𝑦 − 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 → (0,6)

Remember that in Exponential Functions like 𝒇𝒇(𝒙𝒙) = 𝒂𝒂 , the 𝒙𝒙 𝒚𝒚 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 is

always (𝟎𝟎,𝟏𝟏 and in Logarithmic Functions as ) 𝒇𝒇(𝒙𝒙) = 𝐥𝐥𝐥𝐥𝐥𝐥𝒂𝒂 𝒙𝒙 the 𝒙𝒙 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 is

always (𝟏𝟏,𝟎𝟎) In the Basic Sine Function the . 𝒙𝒙 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 points are those whose

𝐬𝐬𝐬𝐬𝐬𝐬𝜽𝜽 = , and the 𝟎𝟎 𝒚𝒚 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 is always (𝟎𝟎,𝟎𝟎) In the Basic Cosine Function the .

𝒙𝒙 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 points are those whose 𝐜𝐜𝐥𝐥𝐬𝐬𝜽𝜽 = , and the 𝟎𝟎 𝒚𝒚 − 𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒄𝒄𝒊𝒊𝒊𝒊𝒊𝒊 is always

(𝟎𝟎,𝟏𝟏) .

MATH VOCABULARY: Intersection Points.

6. VARIATIONS IN A FUNCTION.

INCREASING AND DECREASING. 6.1.

A function is Increasing on an interval 𝒇𝒇 (𝒂𝒂,𝒃𝒃) if for any 𝒙𝒙𝟏𝟏 and 𝒙𝒙 in the 𝟐𝟐

interval such that 𝒙𝒙𝟏𝟏 < 𝒙𝒙𝟐𝟐 then 𝒇𝒇(𝒙𝒙𝟏𝟏) < 𝒇𝒇(𝒙𝒙𝟐𝟐 . Another way to look at this is: as you )

trace the graph from to 𝒂𝒂 (that is from left to right) the graph should go up. 𝒃𝒃

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.30

Unit 09 May

A function is Decreasing on an interval 𝒇𝒇 (𝒂𝒂,𝒃𝒃) if for any 𝒙𝒙𝟏𝟏 and 𝒙𝒙 in the 𝟐𝟐

interval such that 𝒙𝒙𝟏𝟏 < 𝒙𝒙𝟐𝟐 then 𝒇𝒇(𝒙𝒙𝟏𝟏) > 𝒇𝒇(𝒙𝒙𝟐𝟐 . Another way to look at this is: as you )

trace the graph from to 𝒂𝒂 (that is from left to right) the graph should go down. 𝒃𝒃

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.31

Unit 09 May

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.32

Unit 09 May

MAXIMA AND MINIMA. 6.2.

A function has a Relative (or Local) Maximum at a point if its ordinate is 𝒇𝒇

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 is 𝒇𝒇

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 smallest 𝒇𝒇

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.33

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 − 𝒗𝒗𝒂𝒂𝒍𝒍𝒗𝒗𝒊𝒊𝒊𝒊 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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.34

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The only Periodic Functions studied are the Trigonometric Functions seen.

MATH VOCABULARY: Periodic Function, Cycle, Period, Amplitude.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.35

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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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.36

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.

Axel Cotón Gutiérrez Mathematics 4º ESO 4.9.37