I will now explain how to represent a second degree function in the coordinate axes step by step.
To represent a second degree function, you need to know what form the function will take in order to find the key points that will determine the layout of the function.
What are these points and how are they obtained?
We’re going to see it little by little.
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What shape does a second degree function have?
Before you start playing a second-degree role, you need to know what shape it will take.
The second degree functions are in the form of a parabola, which can go with the vertex upwards:
Or with the vertex down:
Whether the parabola is with the vertex up or down depends on the signs of the coefficient “a” of the second degree function:
Any second degree function, whether complete or incomplete, will be in the form of a parabola. It will be different in each case (more open, less open, with the vertex located at a different point…) but it will always be a parable.
Which points are needed to represent a second degree function
Now that you know that the functions of the second degree are in the form of a parable, it will be much easier for you to represent them.
But…. Where do I start to play a second-degree role?
To represent a function on the coordinate axes you need to make a table of values and find a series of points, so that when you join them together later, you can represent that function.
These points are obtained by choosing values of x and calculating the values of y that correspond to them (if you don’t know how to do it, in the example below I will explain it to you).
In the case of second degree functions, we cannot choose any value of x to obtain the points that define the function, as we would be lost and would not know the exact shape of the parabola.
The points of the parabola must be calculated according to a criterion, which allows us to draw the parabola by joining them together.
Therefore, the first point we need is the vertex of the parable.
Once we have that point, we can give it two or three values of x to the right of the vertex and calculate the value of y, so we would have two or three more points to the right of the vertex and do the same for the left.
Finally we will have about 5 or 7 points that will define the parable we are looking for perfectly.
I’ll tell you this so you have a global idea. We’ll take a closer look at it with a step-by-step example.
Parabola vertex coordinates
As I mentioned before, the vertex is the first point you have to get when it comes to representing a second degree function.
And you may wonder how to know what coordinates the vertex has?
Then let’s go see him.
The x-coordinate of the vertex is determined by the formula:
Where a and b are the coefficients of the second degree function:
To calculate the “y” coordinate of the vertex, we only have to substitute the x value obtained in the second degree function and operate.
Let’s look at it with an example: Get the coordinates of the vertex of the next second degree function:
We start by calculating the x-coordinate of the vertex with the formula:
In this second degree function the coefficients a and b are 1 and -1 respectively:
Therefore, we substitute those two values in the formula and operate:
The x-coordinate of the vertex is 0.5.
We are now going to calculate the y-coordinate of the vertex.
In case you didn’t know, the y-coordinate of the vertex is the same as f(x):
So the function can also be expressed as:
In the previous expression, we replace the x with the x-coordinate of the vertex we calculated with the formula:
And we operate:
The y coordinate of the vertex is -2.25.
Therefore, the coordinates of the vertex, which we will call point V, are:
Example of representation of a quadratic function
Let’s represent the second degree function from the example above:
We will continue to obtain points, in a table of values, which we will construct according to a criterion. Finally, we will represent the points obtained and by joining them together, we will have the second degree function represented in the coordinate axes.
Constructing the table of values
A table of values is a table where the points needed to draw a function are collected.
The values of x are chosen and the values of y are calculated, replacing each value of x chosen in the function.
Let’s take it one step at a time for our second grade show.
The first point we have is the vertex of the parable, which we calculated earlier. For the value of x=0.5, we calculated a value of y=-2.25. We put it in our table of values:
Once we have the vertex, we are going to give the x, three values greater than the value x of the vertex, that is to say, that they remain to the right in the number line, as for example 1, 2 and 3:
Now, we calculate the value of “y”, for each value of x chosen, substituting each value in the function:
Now we give the x another three values smaller than the x-value of the vertex, that is to say, that they remain on the left on the number line, such as 0, -1 and -2:
And we calculate the value of’y’ for each of these values of x:
Now we are going to represent all the points on the coordinate axes. We start at the top:
Continue along the three points on your right and the other three on your left:
The shape of the parable can already be sensed.
Finally, we join the points and we will have the function of second degree represented:
Why did we start at the apex and then give points to his left and right?
The parabola is a symmetrical function. When we start at the top, we have the point of reference. When we know that we are drawing a parable we need to know what that parable will be like on both sides of the vertex and that’s why we look for points to the right and to the left.
This method of drawing second degree functions is valid for both complete and incomplete functions. If you notice, we only depend on parameters a and b, which are the ones that mark the vertex.