Function

What is a function?

A function is an equation that describes a specific relationship between XX and YY.
Every time we change XX, we get a different YY.

Linear function –

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

Quadratic function –

Parabola, XX is in the square.

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Test yourself on the quadratic function!

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Look at the linear function represented in the diagram.

When is the function positive?

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Function

What is a function?

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

For example:
Y=x+5Y = x+5

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

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

What is a linear function?

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

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

bb represents the point where the function intersects the YY axis.

Important Points:

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

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

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What is a quadratic 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=ax2+bx+cY = ax^2+bx+c

When -
aa - must be different from 00.

Minimum and maximum parabolas

Minimum Parabola – also called a smiling parabola.
Vertex of the parabola – the point where YY is the smallest.
If aa 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 YY is the highest.
If aa in the equation is negative (sad) – the parabola is a maximum parabola

How do you find the vertex of the parabola?

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

Xvertex=b2aX_{\text{vertex}} = \frac{-b}{2a}

We substitute aa and bb into the formula from the function's equation and find XX.
After finding the vertex XX, we substitute it into the original function's equation and get the vertex YY.

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

The vertex XX we obtained, we will substitute into the original equation to find the value of the vertex YY.

How do you find the intercepts of the axes with the parabola?

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

Finding the intervals of increase and decrease of a quadratic function based on a graph

Intervals of increase and decrease describe the XX 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 XX and what happens when the x-values are greater than the vertex XX.
The following quadratic function is given:
It is given that the vertex of the parabola is (4,1)(-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?
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