A function is an equation that describes a specific relationship between $X$ and $Y$.

Every time we change $X$, we get a different $Y$.

A function is an equation that describes a specific relationship between $X$ and $Y$.

Every time we change $X$, we get a different $Y$.

Looks like a straight line, $X$ is in the first degree.

Parabola, $X$ is in the square.

Look at the linear function represented in the diagram.

When is the function positive?

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:

Test your knowledge

Question 1

Look at the function shown in the figure.

When is the function positive?

Question 2

Solve the following inequality:

\( 5x+8<9 \)

Question 3

Solve the inequality:

\( 5-3x>-10 \)

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

Do you know what the answer is?

Question 1

What is the solution to the following inequality?

\( 10x-4≤-3x-8 \)

Question 2

For the function in front of you, the slope is?

Question 3

For the function in front of you, the slope is?

Related Subjects

- Two linear equations with two unknowns
- Algebraic solution for linear equations with two unknowns
- Substitution method for two linear equations with two unknowns
- Solving with the method of equalization for systems of two linear equations with two unknowns
- Linear equation with two variables
- Solution with graphical method for linear equations with two variables
- Verbal Problem Solving With a System of Linear Equations
- Inequalities
- Inequalities with Absolute Value
- Special Cases of Equations
- Equations with Fractions
- Equation with variable in the denominator
- Absolute Value
- Absolute Value Inequalities
- Quadratice Equations and Systems of Quadraric Equations
- Quadratic Equations System - Algebraic and Graphical Solution
- Solution of a system of equations - one of them is linear and the other quadratic
- Intersection between two parabolas
- Word Problems
- Properties of the roots of quadratic equations - Vieta's formulas
- Ways to represent a quadratic function
- Various Forms of the Quadratic Function
- Standard Form of the Quadratic Function
- Vertex form of the quadratic equation
- Factored form of the quadratic function
- Graphical Representation of a Function that Represents Direct Proportionality
- Slope in the Function y=mx
- Finding a Linear Equation
- Positive and Negativity of a Linear Function
- Representation of Phenomena Using Linear Functions
- The quadratic function
- Quadratic Inequality
- Parabola
- Methods for solving a quadratic function
- Completing the square in a quadratic equation
- Squared Trinomial
- Families of Parabolas
- The functions y=x²
- Family of Parabolas y=x²+c: Vertical Shift
- Family of Parabolas y=(x-p)²
- Family of Parabolas y=(x-p)²+k (combination of horizontal and vertical shifts)