Why were quadratic numbers needed

Why is?
In addition to the linear functions, i.e. the straight lines, the quadratic functions play a major role in mathematics. If you draw such a function in a coordinate system, you get a parabola.

In this lesson we learn what parabolas are all about, to create their table of values ​​and to draw them.
 
In this lesson you will learn
1.Why you need quadratic functions.
2.What constitutes a quadratic function
3.Create the table of values.
4.Draw parabolas.
5.To sketch your course without an invoice.
 Why do you need quadratic functions?
Functions such as y = 2x + 3 or y = x2 - 4, are formulas that can be used to calculate something. In the natural sciences, in craft and technology, but also in the financial world, you have to constantly calculate something, be it prices, required quantities, distances or other important data.

The simplest type of function is linear. You already know them, e.g. y = 2x + 3. Linear functions can be used for problems in which the size that is involved increases or decreases evenly. You can use them to calculate mobile phone costs, purchase prices or travel times at the same speed.
 

     
Other quantities do not behave linearly. A car that is accelerated becomes faster and faster. The distance covered per second grows quadratically. A circle or a spherical surface also becomes quadratically larger with increasing radius.

"Square" means: If you double the radius, the area quadruples! Linear functions (formulas) cannot be used for these and many other problems. Here quadratic functions are popular.
 
The functional equation
Quadratic functions can be recognized by the fact that they have an x2 have the highest power of x.
 






Note:
General form:
y = ax2 + bx + c
Normal parabola:
y = x2
Examples:

y = x2 ; y = 3x2 ; y = -x2 ; y = 2x2 + 4x;
y = -3x2 - 6; y = x2 + 2x - 4

No quadratic functions would be:
y = 4x + 2 (linear); y = x3 + 2x2 - 5x (cubic)
 
In general, quadratic functions have the following form:

y = ax2 + bx + c

Here a, b and c are arbitrary numbers.

The simplest representative is therefore the following function:

y = x2 They are calledNormal parabola.
 Table of values ​​and graph
In order to be able to draw a function in the coordinate system, you have to create a table of values.

To do this, one calculates for different x what comes out in each case.

Example:

For the function y = x2 a table of values ​​for x ϵ [-3; 3] and ∆x = 0.5 and draw its graph. (i.e. from -3 to 3, step size = 0.5)



You can see the result on the right.
 










 
The shape of all parabolas is essentially the same. However, they differ in their curvature. You can therefore only draw some with the template, all the others are drawn by hand.

They also differ in whether they open up or down, and where the vertex is.
 
A second example

For the function y = -0.5x2 + 2x + 1 a table of values ​​for x ϵ [-1; 5] and ∆x = 1 and draw their graph.



On the right you can see the result again.
 The problem with the calculator
When entering the x-values ​​into the calculator, many students have serious problems, especially when it comes to negative x-values. You only have to pay attention to one thing:

→ Always put minus numbers that are squared in brackets !!!

Try the function y = -3x2 - 2x + 1
for x ϵ [-3; -1] and ∆x = 1.

x = -3: Entry: - 3x ( – 3 ) 2 – 2 x – 3 + 1  → -20
x = -2: Entry: - 3x ( – 2 ) 2 – 2 x – 2 + 1  →  -7
x = -1: Entry: - 3x ( – 1 ) 2 – 2 x – 1 + 1  →  0
 

      
Minus numbers,
which are squared, always put in brackets !!!
The meaning of the parameter a
Even without a table of values, a parabola can be roughly sketched if one understands how the parameter a, i.e. the number in front of the x2affects the shape.

In addition to the parameter a, the position of the vertex is also important. But that's only available in the next lesson.
 




Note:
Plus before the x2:
open to the top.
Minus before the x2:
open at the bottom.
 
Examples:

y = 2x2 - 5x + 1open at the top, tight
y = -0.2x2 + x + 4open at the bottom, wide
y = -1.3x2 - 2xopen downwards, tight