Function

A function is an equation that describes a certain relationship between $X$ and $Y$.
In every function, $X$ is the independent variable and $Y$ is the dependent variable. This means that every time we change $X$, we get a different $Y$.
In other words, the $Y$ we get will depend on the $X$ we substitute into the function.
$Y$ depends on $X$ and $X$ does not depend on anything.
An important point: for every $X$, there will be only one $Y$!

For example:
$Y = x+5$

In this function, if we substitute $X$, we get a different $Y$ each time.
Let's check:
Substitute $X=1$
and we get:
$y = 1+5$
$y=6$

Substitute $X = 2$ and we get:
$y=7$

A linear function is a function that looks like a straight line.
It belongs to the family of functions $y = mx+b$ where:

$a$ represents the slope of the function - positive or negative
$m>0$ - ascending line
$m<0$ - descending line

$b$ represents the point where the function intersects the $Y$ axis.

Important Points:

• The function will look like a straight line rising or falling or parallel to the $X$ axis, but never parallel to the $Y$ axis.
• We will look at the line from left to right.

Let's look at an example of a linear function:

A quadratic function is a function where X is squared. The quadratic function is also called a second-degree function and is commonly referred to as a parabola.
The basic quadratic function equation is:
$Y = ax^2+bx+c$

When -
$a$ - must be different from $0$.

Minimum Parabola – also called a smiling parabola.
Vertex of the parabola – the point where $Y$ is the smallest.
If $a$ in the equation is positive (happy) – the parabola is a minimum parabola

Maximum Parabola – A maximum parabola is also called a sad parabola
Vertex of the Parabola – The point where $Y$ is the highest.
If $a$ in the equation is negative (sad) – the parabola is a maximum parabola

Finding the vertex of the parabola -
The first method: Using the vertex formula of the parabola.

$X_{\text{vertex}} = \frac{-b}{2a}$

We substitute $a$ and $b$ into the formula from the function's equation and find $X$.
After finding the vertex $X$, we substitute it into the original function's equation and get the vertex $Y$.

The second method: Using two symmetrical points
The formula to find $X$ vertex using two symmetrical points is:

The vertex $X$ we obtained, we will substitute into the original equation to find the value of the vertex $Y$.

To find the intersection point with the $X$ axis:
Substitute $Y=0$ into the quadratic equation and solve using factoring or the quadratic formula.
To find the intersection point with the $Y$ axis:
Substitute $X=0$ into the quadratic equation and find the solutions.

Intervals of increase and decrease describe the $X$ values where the parabola is increasing and where the parabola is decreasing.

Let's see an example:
We will check what happens when the x-values are less than the vertex $X$ and what happens when the x-values are greater than the vertex $X$.
The following quadratic function is given:
It is given that the vertex of the parabola is $(-4, -1)$ and that the parabola is a minimum parabola.
Solution:
We will draw a sketch according to the data and thus clearly see the increasing and decreasing intervals of the function.

Translation of images:

The value of X at the first point + the value of X at the second point, vertex