In this lesson I’m going to explain how to represent the graph of any function, with an exercise solved step by step.
Normally, the functions that we know how to represent are elementary functions, with the help of a table of values, but to graphically represent any function, which is not elementary, we must analyze a series of characteristics of the function and then represent them in the axes.
We are going to see each one of them at the same time that we are solving an example.
Índice de Contenidos
Graphic representation of functions
We are going to represent the next function:
In order to do this, it is necessary to analyze a series of characteristics of the function, which will give us clues as to how the graphical representation of the function will be:
We calculate the domain of the function:
From where we get that:
And the domain is:
With this we know in which points the function does not exist.
2. Cutting points with axes
To know at what point on the y-axis, cut the graph of the function, make x=0 and calculate the value of the function in x=0:
The function will cut the y-axis at the point (0,2).
To know at which point on the x-axis, cut the graph of the function, make y=0 and solve the equation:
This equation will be 0 when its numerator is 0:
Since there is no solution, the graph does not cut the x-axis at any point.
3. Symmetry and periodicity
The function is par when:
If the function is even, it is symmetrical with respect to the y axis.
The function is odd when:
If a function is odd, it is symmetrical with respect to the origin of coordinates.
We calculate f(-x), substituting x for -x in the function and we have it:
On the other hand we calculate -f(x):
We see that f(-x) is not equal to f(x):
Therefore, it is not even and it is not symmetrical with respect to the y-axis.
f(-x) is also not equal to -f(x):
Therefore, it is neither odd nor symmetrical with respect to the origin of coordinates.
The function is also not periodic, since it is not a trigonometric function.
The asymptotes are straight lines to which the graph of the function gets closer and closer to them, to infinity, but never touches them.
There are three types of asymptotes
There will be a horizontal asymptote in:
When the limit when x tends to infinity is equal to a constant:
Or when the limit when x tends to less infinity is equal to a constant:
Let’s see if our function has horizontal asymptotes.
We calculate the limit of the function when x tends to infinity:
It is not equal to any constant, so with this condition we have not obtained any horizontal asymptote.
We are going to calculate the limit when x tends to less infinity.
The result of which is also not a constant.
Therefore, our function does not have horizontal asymptotes.
There will be a vertical asymptote in:
If the limit for that constant k is equal to infinity or less infinity:
With what constant do we calculate the limit to check if the function has vertical asymptotes?
With the values of x that do not exist in the domain. In our case for x=-1.
Therefore, we calculate the limit of the function when x tends to -1:
Which can be more infinite or less infinite.
We calculate its lateral limits. When x tends to -1 on the left the limit is equal to less infinity:
Y the limit when x tends to -1 on the right equals more infinity.
Therefore in x=-1 we have a vertical asymptote:
We are going to represent in everything that this means. We represent the line x=1 which is the vertical asymptote:
When the function approaches this asymptote on the right, it tends to infinity and when it approaches the asymptote on the left it tends to less infinity.
We represent this with two small strokes of the function as follows:
Oblique asymptotes are only calculated when there are no horizontal asymptotes.
On the other hand, for oblique asymptotes to exist, we must have a rational function and the degree of the numerator must be a degree greater than the degree of the denominator.
They are straight lines that have this formula:
Where m is:
And n is:
Since in our case we did not have any horizontal asymptote, we are going to calculate the oblique asymptote.
We calculate m:
And we calculate n:
We substitute the values of m and n in the formula of the line and we are left:
When we represent it in the axes it looks like this:
5. Growth and decrease intervals
Now we are going to calculate the growth and decrease intervals of the graph.
As I explained in the lesson the intervals of growth and decrease of a function:
So, let’s calculate the derivative of the function:
And we are going to study its sign.
For it, we must find the values that make 0 the function, from where also we will obtain the intervals.
We equate the function to zero and solve the equation:
Whose solutions are:
These two solutions are placed on the number line, along with the values of x that do not belong to the domain, that is, we also include -1:
Now we study its sign, calculating the value of f'(x) for one point of each interval and we are left:
6. Maximum minima and tipping points (relative extremes)
The possible relative extremes are those that make the value of the first derivative of the function zero, as I explained in the lesson on calculating maxima, minima and inflection points:
We have calculated these points in the previous section and they are:
We must calculate the value of the second derivative of the function at these two points to check if they are maximums, minimums or inflection points:
We calculate the second derivative of the function:
And we check the sign of the second derivative for x=0
The value of the second derivative for x=0 is 2, which is greater than 0, so in x=0 there is a minimum:
We do the same for x=-2:
The value of the second derivative for x=-2 is -2, which is less than 0, so in x=0 there is a maximum:
We have not come up with any inflection points.
We are going to calculate the points that would be possible inflection points, equating the second derivative to 0:
That has no solution, so there is no value that makes the second derivative is zero and therefore, the function has no inflection points.
7. Concavity and convexity
Finally, we will obtain in which intervals the function is concave and which intervals are convex.
To do this, we will study the sign of the second derivative of the function:
To find the intervals, we equal the second derivative to 0, which we have already obtained in the previous section that has no solution.
Therefore, since we do not have solutions obtained by equalizing the second derivative to zero, in the numerical line we place the values that do not belong to the domain, that is, the point x=-1:
Now we study its sign, calculating the value of f”(x) for one point of each interval and we are left:
We already have all the necessary data to be able to graph the function.
We draw the relative extremes and join the lines of graphs, knowing that they cannot touch the asymptotes and knowing the sections in which they are increasing and decreasing. We will be left with a graph like this one:
We also check how the section that goes from less infinite to infinite the function is concave and in the other interval is convex.