In this explainer, we will learn how to determine the sign of a function from its equation or graph.
Definition: Sign of a Function
The sign of a function is a description indicating whether the function is positive, negative, or zero. For the function on an interval ,
- the sign is positive if for all in ,
- the sign is negative if for all in .
The sign of the function is zero for those values of where . Just as the number 0 is neither positive nor negative, the sign of is zero when is neither positive nor negative.
Letβs consider three types of functions. The first is a constant function in the form , where is a real number. The second is a linear function in the form , where and are real numbers, with representing the functionβs slope and representing its -intercept. The third is a quadratic function in the form , where , , and are real numbers, and is not equal to 0. Examples of each of these types of functions and their graphs are shown below.
We can determine the sign or signs of all of these functions by analyzing the functionsβ graphs.
Property: Relationship between the Sign of a Function and Its Graph
- When the graph of a function is above the -axis, the functionβs sign is positive.
- When the graph of a function is below the -axis, the functionβs sign is negative.
- At any -intercepts of the graph of a function, the functionβs sign is equal to zero.
We can see that the graph of the constant function is entirely above the -axis, and the arrows tell us that it extends infinitely to both the left and the right. This means it will never intersect or be below the -axis. Thus, we say this function is positive for all real numbers .
Next, we can see that the graph of the linear function is below the -axis for some values of and above the -axis for others. We can also see that it intersects the -axis once. To determine the values of for which the function is positive, negative, and zero, we can find the x-intercept of its graph by substituting 0 for and then solving for as follows:
Since the graph intersects the -axis at , we know that the function is positive for all real numbers such that and negative for all real numbers such that . We also know that the functionβs sign is zero when .
Finally, we can see that the graph of the quadratic function is below the -axis for some values of and above the -axis for others. We can also see that the graph intersects the -axis twice, at both and , so the quadratic function has two distinct real roots. Therefore, we know that the function is positive for all real numbers , such that or , and that it is negative for all real numbers , such that . We also know that the functionβs sign is zero when and .
Properties: Signs of Constant, Linear, and Quadratic Functions
- A constant function in the form can only be positive, negative, or zero. It cannot have different signs within different intervals.
- A linear function in the form is always positive, negative, and zero for different values of
when is not equal to 0.
- When , its sign is the opposite of that of .
- When , its sign is the same as that of .
- When , its sign is zero.
- A quadratic function in the form with two distinct real roots is always positive, negative,
and zero for different values of .
- When is less than the smaller root or greater than the larger root, its sign is the same as that of .
- When is between the roots, its sign is the opposite of that of .
- At the roots, its sign is zero.
Now, letβs look at some examples of these types of functions and how to determine their signs by graphing them.
Example 1: Determining the Sign of a Constant Function
In which of the following intervals is negative?
Answer
Recall that the sign of a function can be positive, negative, or equal to zero. It is positive in an interval in which its graph is above the -axis on a coordinate plane, negative in an interval in which its graph is below the -axis, and zero at the -intercepts of the graph.
To help determine the interval in which is negative, letβs begin by graphing on a coordinate plane. Recall that the graph of a function in the form , where is a constant, is a horizontal line. This is because no matter what value of we input into the function, we will always get the same output value. In this case, the output value will always be , so our graph will appear as follows:
We can see that the graph is entirely below the -axis and that inputting any real-number value of into the function will always give us . This means the graph will never intersect or be above the -axis. In other words, the sign of the function will never be zero or positive, so it must always be negative. Thus, the interval in which the function is negative is .
In the following problem, we will learn how to determine the sign of a linear function.
Example 2: Determining the Sign of a Linear Function over Different Intervals
Determine the sign of the function .
Answer
Recall that the sign of a function is a description indicating whether the function is positive, negative, or zero. We can determine a functionβs sign graphically. We know that the sign is positive in an interval in which the functionβs graph is above the -axis, zero at the -intercepts of its graph, and negative in an interval in which its graph is below the -axis.
First, letβs determine the -intercept of the functionβs graph by setting equal to 0 and solving for :
This tells us that the graph intersects the -axis at the point . From the functionβs rule, we are also able to determine that the -intercept of the graph is 5, so by drawing a line through point and point , we can construct the graph of as shown:
We can see that the graph is above the -axis for all real-number values of less than 1, that it intersects the -axis at 1, and that it is below the -axis for all real-number values of greater than 1. Thus, we can conclude that the function is positive when , it is negative when , and it equals zero when .
Note
By inputting values of into our function and observing the signs of the resulting output values, we may be able to detect possible errors. Letβs input some values of that are less than 1 and some that are greater than 1, as well as the value of 1 itself:
Notice that input values less than 1 return output values greater than 0 and that input values greater than 1 return output values less than 0. Inputting 1 itself returns a value of 0. This is consistent with what we would expect.
Note that, in the problem we just solved, the function is in the form . Remember that the sign of a linear function in this form can also be determined algebraically.
- When , its sign is the opposite of that of .
- When , its sign is the same as that of .
- When , its sign is zero.
In this case, and , so the value of is , or 1. Thus, since the sign of is negative, we know that the function is positive when , it is negative when , and it is zero when . This is the same answer we got when graphing the function.
Next, we will graph a quadratic function to help determine its sign over different intervals.
Example 3: Determining the Sign of a Quadratic Function over Different Intervals
Determine the sign of the function .
Answer
In this problem, we are given the quadratic function . We know that for values of where , its sign is positive; for values of where , its sign is negative; and for values of where , its sign is equal to zero. Letβs start by finding the values of for which the sign of is zero. Setting equal to 0 gives us
To solve this equation for , we should check to see if we can factor the left side into a pair of binomial expressions. If we can, we know that the first term in each of the factors will be , since the product of and is . We also know that the second terms will have to have a product of 16 and a sum of 10. Since and , we can factor the left side to get
Since the product of the two factors is equal to 0, one of the two factors must have a value of 0. That is, either or . We can solve the first equation for by subtracting 8 from both sides and the second equation by subtracting 2 from both sides. This tells us that either or . In other words, the zeros of the function are and .
We can determine the sign of a function from its graph, so letβs sketch a graph of . Since the coefficient of the -term is positive, we know that the graph is a parabola that opens upward. We have already shown that the graphβs -intercepts are and , and since we know that the graphβs -intercept is 16. Thus, our graph should appear roughly as follows:
We can see that the graph is above the -axis for all values of less than and also those greater than , that it intersects the -axis at and , and that it is below the -axis for all values of between and . This means that the function is positive when and , it is negative when , and it equals zero when and . Using set notation, we would say that the function is positive when , it is negative when , and it equals zero when .
Note that, in the problem we just solved, the function is in the form , and it has two distinct roots. Remember that the sign of such a quadratic function can also be determined algebraically.
- When is less than the smaller root or greater than the larger root, its sign is the same as that of .
- When is between the roots, its sign is the opposite of that of .
- At the roots, its sign is zero.
In this case, , and the roots of the function are and . Since the sign of is positive, we know that the function is positive when and , it is negative when , and it is zero when and when . This is the same answer we got when graphing the function.
Also note that, in the problem we just solved, we were able to factor the left side of the equation . This allowed us to determine that the corresponding quadratic function had two distinct real roots. However, this will not always be the case.
Consider the quadratic function . Setting equal to 0 gives us , but there is no apparent way to factor the left side of the equation. We can confirm that the left side cannot be factored by finding the discriminant of the equation. For a quadratic equation in the form , the discriminant, , is equal to . Thus, the discriminant for the equation is
Since the discriminant is negative, we know that the equation has no real solutions and, therefore, that the function has no real roots. Since the functionβs leading coefficient is positive, we also know that the functionβs graph is a parabola that opens upward, so the graph will appear roughly as follows:
Since the graph is entirely above the -axis, the function is positive for all real values of .
Property: Relationship between the Discriminant of a Quadratic Equation and the Sign of the Corresponding Quadratic Function π(π₯) = ππ₯2 + ππ₯ + π
- When the discriminant of a quadratic equation is negative, the corresponding function has no real roots. The functionβs sign will always be the same as the sign of .
- When the discriminant of a quadratic equation is zero, the corresponding function has one real root. The functionβs sign will always be zero at the root and the same as that of for all other real values of .
- When the discriminant of a quadratic equation is positive, the corresponding function has two real roots. The functionβs sign will always be the same as that of when is less than the smaller root or greater than the larger root, the opposite of that of when is between the roots, and zero at the roots.
In the example that follows, we will look for the values of for which the sign of a linear function and the sign of a quadratic function are both positive.
Example 4: Determining an Interval Where a Linear and a Quadratic Function Share the Same Sign
What are the values of for which the functions and are both positive?
Answer
In this problem, we are asked for the values of for which two functions are both positive. Recall that positive is one of the possible signs of a function. To begin, letβs consider the function . We know that it is positive for any value of where , so we can write this as the inequality
Adding 5 to both sides gives us , which can be written in interval notation as . That is, the function is positive for all values of greater than 5.
Now, letβs look at the function . First, we will determine where has a sign of zero. We will do this by setting equal to 0, giving us the equation
We should now check to see if we can factor the left side of this equation into a pair of binomial expressions to solve the equation for . Since the product of and is , we know that if we can, the first term in each of the factors will be . We also know that the second terms will have to have a product of and a sum of 2. Since and , we can factor the left side to get
The product of the two factors is equal to 0, so one of the two factors must have a value of 0. That is, either or . We can solve the first equation by adding 6 to both sides, and we can solve the second by subtracting 8 from both sides. This tells us that either or , so the zeros of the function are 6 and .
We can find the sign of a function graphically, so letβs sketch a graph of . The coefficient of the -term is positive, so we know that the graph is a parabola that opens upward. We have already shown that the -intercepts of the graph are 6 and , and since we know that the -intercept is . Thus, our graph should be similar to the one below:
We can see that the graph is above the -axis for all values of less than and also those greater than 6, so the function is positive when and also when . In interval notation, this can be written as .
Now that we know that is positive when and that is positive when or , we can determine the values of for which both functions are positive. Since any value of less than is not also greater than 5, we can ignore the interval and determine only the values of that are both greater than 5 and greater than 6. The values of greater than both 5 and 6 are just those greater than 6, so we know that the values of for which the functions and are both positive are those that satisfy the inequality .
As a final example, weβll determine the interval in which the sign of a quadratic function and the sign of another quadratic function are both negative.
Example 5: Determining an Interval Where Two Quadratic Functions Share the Same Sign
Determine the interval where the sign of both of the two functions and is negative in .
Answer
In this problem, we are asked to find the interval where the signs of two functions are both negative. Recall that the sign of a function is negative on an interval if the value of the function is less than 0 on that interval.
We can determine the sign of a function graphically, and to sketch the graph of a quadratic function, we need to determine its -intercepts. First, letβs consider the function . To find the -intercepts of this functionβs graph, we can begin by setting equal to 0. This gives us the equation
To solve this equation for , we should check to see if we can factor the left side into a pair of binomial expressions. If we can, we know that the first terms in the factors will be and , since the product of and is . We also know that the second terms will have to have a product of . Since , we can try to factor the left side as , giving us the equation
Since the product of and is , we know that we have factored correctly. We can see that the product of the two factors is equal to 0, so one of the two factors must have a value of 0. That is, either or . Solving these equations for , we get and
This tells us that either or , so the zeros of the function are and 6.
Now, letβs sketch a graph of . Since the coefficient of the -term is positive, we know that the graph is a parabola that opens upward. We have already shown that the graphβs -intercepts are and 6, and since we know that its -intercept is . Thus, our graph should appear roughly as follows:
We can see that the graph is below the -axis for all values of greater than and less than 6. This means that the function is negative when is between and 6. In interval notation, this can be written as .
Next, letβs consider the function . Setting equal to 0 gives us the equation
To solve this equation for , we must again check to see if we can factor the left side into a pair of binomial expressions. If we can, we know that the first term in each of the factors will be , since the product of and is . We also know that the second terms will have to have a product of and a sum of . Since and , we can factor the left side to get
Since the product of the two factors is equal to 0, one of the two factors must again have a value of 0. That is, either or Solving these equations for , we get and
Now, we can sketch a graph of . The coefficient of the -term is positive, so we again know that the graph is a parabola that opens upward. We have already shown that the -intercepts of the graph are 5 and , and since we know that the -intercept is . Thus, our graph should be similar to the one below:
This time, we can see that the graph is below the -axis for all values of greater than and less than 5, so the function is negative when and . In interval notation, this can be written as .
Now that we know that is negative when is in the interval and that is negative when is in the interval , we can determine the interval in which both functions are negative. Since the interval is entirely within the interval , or the interval , all values of within the interval would also be within the interval.
Thus, we know that the values of for which the functions and are both negative are within the interval .
Note
This can be demonstrated graphically by sketching and on the same coordinate plane as shown.
Now letβs finish by recapping some key points.
Key Points
- A constant function is either positive, negative, or zero for all real values of .
- A linear function in the form , where
, always has an interval in which it is negative,
an interval in which it is positive, and an -intercept where its sign is zero.
- When , its sign is the opposite of that of .
- When , its sign is the same as that of .
- When , its sign is zero.
- When the discriminant of a quadratic equation is negative, the corresponding function in the form has no real roots. The functionβs sign is always the same as the sign of .
- When the discriminant of a quadratic equation is zero, the corresponding function in the form has one real root. The functionβs sign is always zero at the root and the same as that of for all other real values of .
- When the discriminant of a quadratic equation is positive, the corresponding function in the form has two real roots. The functionβs sign is always the same as that of when is less than the smaller root or greater than the larger root, the opposite of that of when is between the roots, and zero at the roots.
- To determine the sign of a function in different intervals, it is often helpful to construct the functionβs graph. When the graph is above the -axis, the sign of the function is positive; when it is below the -axis, the sign of the function is negative; and at its -intercepts, the sign of the function is equal to zero.
