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Quadratic function
A function of second degree is written as
Explanation
You can factorize a second degree function. Often you can find the solution with little effort, because usually applies a = 1. That you can see clearly
If the solution is not easy you can use a formula. That we are going to develop here, and start with the general form
We separate a with brackets
For the square we are going to prepare a binomial formula
and write this as
The term with an x is handled seperately
Here you can find two solutions

and 


and 


and 
Alternative
If the value of b is large, you can develop alternative formulas, and then you get
and
If you see this it all looks pretty simple. Now we can also calculate
and
That we knew of course already. With this we can calculate the x coordinate of the top, because that is exactly in the middle between x_{1} and x_{2}.
Example 1
We will use the quadratic formule for
This gives (x − 3)(x + 5) = 0 but we could have found this by trial and error. The alternative formulas give
In addition, we see that
or, using the formula
or, using the formula
from that follws
There are many different ways to calculate the coordinates of the top. It depends on how far you are already advanced in mathematics. To complete the picture is here the explanation.
Example 2
In the example y = x^{2} + 2x + 15 you find the top by calculating the lowest value of the function. You can do that by serating a square
A square is always positive, and the lowest value is 0. That occurs at x = –1 and then y = –16. You see that right away.
Example 3
You can also apply differentiation. With it you can determine the rate of rise of a continuous function. This is the tangent. If that is exactly horizontal, you have the top found. We take again the function
and determine the first derivative
The tangent has the value 0 in the top, so
This is by far the easiest way.