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.

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

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

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

Question 2

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

Question 3

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

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

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

Question 2

Look at the linear function represented in the diagram.

When is the function positive?

Question 3

Look at the function shown in the figure.

When is the function positive?

Look at the linear function represented in the diagram.

When is the function positive?

The function is positive when it is above the X-axis.

Let's note that the intersection point of the graph with the X-axis is:

$(2,0)$ meaning any number greater than 2:

x > 2

x>2

Look at the function shown in the figure.

When is the function positive?

The function we see is a decreasing function,

Because as X increases, the value of Y decreases, creating the slope of the function.

We know that this function intersects the X-axis at the point x=-4

Therefore, we can understand that up to -4, the values of Y are greater than 0, and after -4, the values of Y are less than zero.

Therefore, the function will be positive only when

X < -4

-4 > x

Solve the following inequality:

5x+8<9

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

x<\frac{1}{5}

Solve the inequality:

5-3x>-10

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

5 > x

What is the solution to the following inequality?

$10x-4≤-3x-8$

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

$10x-4 ≤ -3x-8$

We start by organizing the sections:

$10x+3x-4 ≤ -8$

$13x-4 ≤ -8$

$13x ≤ -4$

Divide by 13 to isolate the X

$x≤-\frac{4}{13}$

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than$-\frac{4}{13}$, although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to$-\frac{4}{13}$, and only smaller than it. We know it must be large and equal, so this answer is rejected.

Therefore, answer B is the correct one!

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
- Graphs of Direct Proportionality Functions
- 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
- The quadratic equation
- 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)