Maybe you know this situation: You are just about to think about how much you have already spent this month. There were the drinks from the last cozy barbecue party, the costs for the canteen and your mobile phone tariff. On the other hand, you can record a small plus from the monthly pocket money and your little job in the café. You draw yourself a coordinate system with the x-axis indicating that you have made no profit or loss in a period. If you want to know on which day you will make a loss or a profit, knowledge about zeros is important for you.
You can determine zeros for a function, regardless of whether it is a linear function, quadratic function or function with a root. They are also often part of a curve discussion. On the one hand you can use it to determine the point of intersection with the x-axis, but also the maximum or minimum of a derivative function. Therefore, this explanation should revolve around the following: Calculate zeros. Have fun!
Calculating zeros – basic knowledge
The zero is one of the most important intersections of a function. In addition to the y-intercept, it is one of the Intersections with the coordinate axes. The zero is part of the curve discussion. This helps with a better idea of the function graph in order to be able to draw the function later in a coordinate system and is also an important computational tool in mathematics.
A root of a function is a number a from the domain of the function for which .
Graphically, the root designates the x-value of the point of intersection or point of contact of a function f with the x-axis.
So you’re looking for him x value of the function f, for which becomes .
There also as y can be written, the root is the point at which is, which is on the x-axis.
Therefore you can also use the zeros as points on the x-axis read from a recorded function.
The function f intersects the x-axis at point S(2|0).
However, since intersections with the x-axis always have the y-value 0, you are only interested in the x-value.
this one x-value of the intersection with the x-axis can you then as zero describe.
The zero of the function f is therefore at .
Calculate Roots – Intersection or Touch Point
Zeros can be points of intersection or points of contact. The names already tell you what the function graphs look like at the zero point:
- zeros as a intersection are a point where the function graph crosses the x-axis cuts.
- zeros as touchpoint are a point where the function graph has the x-axis only touched. This graph remains positive or negative and also has the at the point of contact slope 0.
- zeros as saddle point are a point where the function graph crosses the x-axis cuts. The difference to an intersection, however, is that the derivation of the function at this point also includes the value 0 Has.
In the following, you are given zeros for three functions. It is a linear function with an intersection point with the x-axis, a quadratic and a fully rational function with a saddle point.
Figure 1: Intersections for linear, quadratic and integral functions
If you would like to find out more about zeros and the intersections with the coordinate axes, then please have a look at the articles Intersections with the coordinate axes and zero over. If you want to get an overview of all calculations with functions, you can also read the explanation curve discussion look at.
Calculate Roots – Number of roots of a function
There is a rule you can use to remember the maximum number of zeros a function has:
A function has a maximum n zeroswhereby n is the highest exponent of the variable of the function.
So a function can maximum n zeros (but also fewer).
The trigonometric functions are an exception. They can also have an infinite number of zeros.
So a function can one, several or even infinitely many have zeros. That depends entirely on the function type. There are even functions whose graph doesn’t touch the x-axis at all, so it no there is zero.
Calculate roots – multiplicity of roots
In addition, can ea single zero also have very special properties: In addition to the simple zeros, which is always the case for linear functions, there are also double, triple, quadruple, or a total of n-fold zeros.
For example, a double root is an x-value for which a squared factor (such as ) becomes zero. The function shown in the previous image, namely the parabola, also has a double zero at the x-value 10.
You can also recognize the multiple zeros by their appearance. In the following picture there are two different quadratic functions with the multiplicity 2 and 4. The function with the 4-fold zero increases less close to the zero, but then all the more.
Figure 2: Multiplicity of two quadratic functions
Warning: the multiplicity of a single zero has nothing to do with the number of zeros in a function. You are also welcome to read the explanation multiplicity of zeros through to familiarize yourself with the zeros for quadratic functions and also for integral functions.
Calculate Roots – Root and y-intercept
The zero point and the y-intercept belong to the intersection points with the coordinate axes.
As you already know, the zero the Intersection with the x-axis. Of the y-intercept against is the Intersection with the y-axis.
zero
y-intercept
A function can have zero, one, more, or an infinite number of roots (see the rule on the number of roots).
There is only a maximum a y-intercept (regardless of the function).
At the intersection with the x-axis, the y-value is always zero.
At the y-intercept, the x-value is always zero.
By definition, any x-value in a function is exact a assigned a specific y-value. For the x-value x=0 (y-axis) it can only be a give y value. As already mentioned, you can find more information under Intersections with the coordinate axes.
Calculate zeros – formula
To calculate zeros, use this definition:
At the root, the function value is always null ().
To calculate the zero, you need the functional equation in the form .
Here, too, you proceed according to a scheme:
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You put a 0 in the place of the function equation.
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Then rearrange the equation for x so that you get a solution of the form
Although the procedure always follows the same pattern, there are special features and other things to consider for each function type.
Calculate zeros – integral functions
The concept of the fully rational function includes, among other things, the linear and quadratic functions. For values greater than 2, you speak of fully rational functions from degree 3.
Examples of well-known integral functions are the linear function or the quadratic function.
In the following, use examples to see how you calculate the zeros for the different fully rational functions.
Calculate Roots – Linear Function
Linear functions are one of the most elementary functions and are to be dealt with in the application for calculating the zeros by means of equivalent transformations.
A linear function is a function of the form .
Finding the zeros for linear functions is achieved by setting the equation to zero. The x-coordinate of the point of intersection with the x-axis is your zero point. This procedure will be explained to you in the following for linear functions.
Look at the function.
Step 1:
Put a 0 instead of f(x).
Step 2:
Solve for x.
Thus the zero of this linear function is at .
Figure 3: Zero for linear function
The graph of a linear function usually has exactly one root, except for constant functions like y = 2. They are parallels to the x-axis and do not intersect with the x-axis, and therefore have no root.
Exception: The constant function f(x)=0 is identical to the x-axis and has infinitely many points of intersection.
You can find more information on the linear function and how to use a drawing to create the equation of a straight line at linear functions and set up the equation of a straight line.
Calculate Roots – Quadratic Function (Parabola)
In order to calculate a quadratic function, you need a little prior knowledge of any known calculation steps of the midnight formula or the pq formula.
A quadratic function is a function of the form .
There are different ways to calculate the zeros. The procedure depends on what the function term looks like.
Would you like to learn general information about drawing parabolas and also about calculating a vertex? Then take a look at the following pages: parables, Calculate vertex.
Calculate zeros – midnight formula
If the equation the general shape () owns, then you turn the midnight formula to calculate the zero.
For the midnight formula or solution formula, the following formula from the specifications of an equation in the general form applies:
The midnight formula can help you calculate the zeros for the general form.
You are given a quadratic function below. The general form is also given as a support. Use the color coding to orientate yourself a little:
Substitute the values for the coefficients a, b, and c into the midnight formula:
Simplify the expression.
Calculate the term first as an addition and then as a subtraction.
So that means the zeros are at -1 and 1.5.
Figure 4: Roots for quadratic function via midnight formula
Calculate roots – pq formula
The equation can also be in the normal form present, so it comes before the none number a more ().
In this case you use the to solve the zeros pq formula.
The pq formula can be calculated using a normal form in this way:
There is also a separate article on this topic, which you are welcome to read for a better understanding.
You are given a function in normal form below.
Calculation of the zero
Step 1:
Set f(x) equal to zero.
Step 2:
Plug the appropriate numbers for p and q into the pq formula with the correct signs.
Step 3:
Complete the math as much as possible.
Step 4:
So the p/q formula helped you calculate the zeros.
Further information on these calculations can be found under the respective explanations: pq formula , midnight formula.
Calculate roots – 3rd degree function and higher
When solving integral functions of degree 3 or higher, there is no longer a solution formula to calculate the zeros. So there are a few little tricks you can use to come up with a solution for this type of function.
Calculate zeros by factoring out
If the 3rd degree function…