Video: Sign of a Function

In this video, we will learn how to determine the sign of a function from its equation or graph.

17:56

Video Transcript

In this video, we will learn how to determine the sign of a function from its equation or graph. We will begin by defining what we mean by the sign of a function. The sign of any function can be positive, negative, or equal to zero. If a function is positive, it is greater than zero and if it is negative, it is less than zero. Some functions might be more than one of these. They might be positive, negative, and equal to zero for different intervals of the function. Let’s consider some different types of graphs.

We begin with three linear or straight-line graphs. The first graph is a horizontal line and will be of the form 𝑦 equals some constant π‘Ž. As the line is below the π‘₯-axis, this function will always be negative. Our second graph is a vertical line, this time of the form π‘₯ equals some constant π‘Ž. This function will be equal to zero at the point the line crosses the π‘₯-axis. It will be negative for all points below this and positive for all points above. Our third linear function is of the form 𝑦 equals π‘šπ‘₯ plus 𝑏, where π‘š is the slope or gradient and 𝑏 is the 𝑦-intercept. Once again, this function will have one value where it is equal to zero. Part of the function will also be positive, and part of it will be negative. The bit above the π‘₯-axis is positive and the bit below is negative. We can calculate the value at which the function is equal to zero by setting 𝑦 equal to zero. We can then solve this equation to calculate a value of π‘₯.

Let’s now consider what happens when we have a quadratic or cubic function. A quadratic function of the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 will be either u-shaped or n-shaped depending on the sign of the leading coefficient π‘Ž. We will once again be able to calculate the values where the function is equal to zero by setting π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 equal to zero. In our diagram, the function will be positive to the right-hand side of one solution and to the left of the other. The function will be negative between the two values. Whilst it is not true for all quadratic functions, this function is positive, negative, and equal to zero for different values.

The same is true for the cubic function shown. This is equal to zero at three points on the graph. It is positive between our first two solutions and when greater than the third solution. The function is negative when it is less than our first solution or between our second and third solutions. In this video, we will focus on questions involving constant functions, linear functions, and quadratic functions.

In which of the following intervals is 𝑓 of π‘₯ equal to negative eight negative? Is it (A) the open interval from negative ∞ to eight? (B) The open interval from negative eight to ∞. (C) The open interval from negative eight to eight. (D) The open interval from eight to ∞. Or (E) the open interval from negative ∞ to ∞.

Let’s begin by considering what the function 𝑓 of π‘₯ equals negative eight looks like. If we consider the normal coordinate axes, our horizontal axis is the π‘₯-axis and our vertical one is the 𝑦-axis. The 𝑦-axis can be replaced in this case with 𝑓 of π‘₯. We are told that 𝑓 of π‘₯ is equal to negative eight. Therefore, we need to find negative eight on the 𝑦- or 𝑓 of π‘₯ axis. Our function is a horizontal line through this point. This line will continue indefinitely to the left and to the right. As the line is entirely below the π‘₯-axis, it will always be negative. As 𝑓 of π‘₯ equal to negative eight is always negative, the correct answer is option (E) the open interval negative ∞ to ∞. If any function is equal to a constant, it will always just have one sign. In this case, the function is always negative.

In our next question, we will look at a function in the form 𝑦 equals π‘šπ‘₯ plus 𝑏.

Determine the sign of the function 𝑓 of π‘₯ is equal to negative five π‘₯ plus five.

We know that this function is linear as it is of the form 𝑦 equals π‘šπ‘₯ plus 𝑏, where π‘š is the gradient or slope and 𝑏 is the 𝑦-intercept. In our question, the slope of the function is negative five and the 𝑦-intercept is five. As the slope of the function is negative, our graph will slope down to the right. In order to determine the sign of the function, we need to find out where the graph is positive, negative, and also equal to zero. We can see that the graph crosses the π‘₯-axis at one point. This will be the point where the function is equal to zero. When the graph is above the π‘₯-axis, the function will be positive, and when it is below the π‘₯-axis, it will be negative.

To calculate the point at which the function is equal to zero, we will set 𝑓 of π‘₯ equal to zero. Adding five π‘₯ to both sides of this equation gives us five π‘₯ is equal to five. We can then divide both sides of this equation by five, giving us π‘₯ is equal to one. Our function is positive for all π‘₯-values less than one. The function is negative or below the π‘₯-axis for all π‘₯-values greater than one. We can therefore conclude the following. The function is positive when π‘₯ is less than one, the function is negative when π‘₯ is greater than one, and, finally, the function equals zero when π‘₯ equals one. The function 𝑓 of π‘₯ equals negative five π‘₯ plus five is positive, negative, and equals zero for different values of π‘₯.

In our next question, we will look at a quadratic function.

Determine the sign of the function 𝑓 of π‘₯ is equal to π‘₯ squared plus 10π‘₯ plus 16.

This function is quadratic and as the coefficient of π‘₯ squared is positive, it will be u-shaped. In order to determine the sign of any function, we need to work out values where 𝑓 of π‘₯ is positive, negative, and also equal to zero. We can begin by calculating the zeroes of the function by setting 𝑓 of π‘₯ equal to zero. This gives us π‘₯ squared plus 10π‘₯ plus 16 equals zero. The quadratic can be factored into two pairs of parentheses or brackets. The first term in each set of parentheses will be π‘₯ as π‘₯ multiplied by π‘₯ is π‘₯ squared. The second terms in our parentheses need to have a product of 16 and a sum of 10. Eight multiplied by two is equal to 16 and eight plus two is equal to 10. π‘₯ squared plus 10π‘₯ plus 16 factorized is equal to π‘₯ plus eight multiplied by π‘₯ plus two.

As multiplying these two parentheses gives us zero, one of the parentheses themselves must also be equal to zero. Either π‘₯ plus eight equals zero or π‘₯ plus two equals zero. Subtracting eight from both sides of the first equation gives us π‘₯ equals negative eight. Subtracting two from both sides of the second equation gives us π‘₯ is equal to negative two. This means that the function 𝑓 of π‘₯ equal to π‘₯ squared plus 10π‘₯ plus 16 is equal to zero when π‘₯ equals negative eight or π‘₯ equals negative two.

It is now worth sketching the graph 𝑦 equals π‘₯ squared plus 10π‘₯ plus 16. We know that the graph is u-shaped and crosses the π‘₯-axis when π‘₯ equals negative eight and when π‘₯ equals negative two. We also know it crosses the 𝑦-axis when 𝑦 is equal to 16. The function is negative when it is below the π‘₯-axis. This occurs between the values negative eight and negative two. We can write this as an inequality. The function is negative when π‘₯ is greater than negative eight, but less than negative two. The graph is positive when it is above the π‘₯-axis. This occurs when π‘₯ is less than negative eight or when π‘₯ is greater than negative two. We can now write all of this information using interval and set notation.

The function 𝑓 of π‘₯ is positive for any real value apart from those in the closed interval negative eight to negative two. This means that the function is positive for all values apart from those between negative eight and negative two inclusive. The function is negative when π‘₯ exists in the open interval negative eight, negative two. This means that it is negative for any value between negative eight and negative two not including those values. The function is equal to zero when π‘₯ exists in the set of numbers negative eight, negative two. This means that it equals zero only at the two values π‘₯ equals negative eight and π‘₯ equals negative two. We can therefore see that the function 𝑓 of π‘₯ is equal to π‘₯ squared plus 10π‘₯ plus 16 is positive, negative, and equal zero for different values of π‘₯.

In our final question, we will consider two different functions.

What are the values of π‘₯ for which the functions 𝑓 of π‘₯ is equal to π‘₯ minus five and 𝑔 of π‘₯ is equal to π‘₯ squared plus two π‘₯ minus 48 are both positive?

Let’s begin by considering the function 𝑓 of π‘₯ is equal to π‘₯ minus five. If we want this to be positive, 𝑓 of π‘₯ must be greater than zero. This gives us π‘₯ minus five is greater than zero. Adding five to both sides of this inequality gives us π‘₯ is greater than five. 𝑓 of π‘₯ is therefore positive on the open interval five to ∞. It is a positive function for any value greater than five. We will now repeat this process for 𝑔 of π‘₯. This gives us π‘₯ squared plus two π‘₯ minus 48 is greater than zero. To solve any quadratic inequality of this form, we firstly need to find the zeros by setting our function equal to zero. π‘₯ squared plus two π‘₯ minus 48 equals zero.

This can be factored or factorized into two sets of parentheses or brackets. The first term in each bracket is π‘₯. The second terms need to have a product of negative 48 and a sum of two. Six multiplied by eight is equal to 48. This means that negative six multiplied by eight is equal to negative 48. Negative six plus eight is equal to two. Our two sets of parentheses are π‘₯ minus six and π‘₯ plus eight. As the product of these two terms is equal to zero, either π‘₯ minus six equals zero or π‘₯ plus eight is equal to zero. Adding six to both sides of the first equation gives us π‘₯ is equal to six. And subtracting eight from both sides of the second equation gives us π‘₯ is equal to negative eight. This means that the function 𝑔 of π‘₯ is equal to zero when π‘₯ equals six and π‘₯ equals negative eight.

As our function is quadratic and the coefficient of π‘₯ squared is positive, the graph will be u-shaped. This means that it is positive on two sections, when π‘₯ is greater than six and when π‘₯ is less than negative eight. The solution to the inequality π‘₯ squared plus two π‘₯ minus 48 is greater than nought is π‘₯ is less than negative eight or π‘₯ is greater than six. This can also be written using interval notation. 𝑔 of π‘₯ is positive in the open interval negative ∞ to negative eight or the open interval six to ∞.

We want to work out the values of π‘₯ where both functions are positive. Let’s consider a number line with the key values five, negative eight, and six marked on. We know that 𝑓 of π‘₯ is positive for all values greater than five. 𝑔 of π‘₯ is positive for all values less than negative eight and greater than six. This means that both functions are positive when π‘₯ is greater than six. This could also be written using interval notation as the open interval six to ∞.

We will now summarize the key points from this video. A constant function of the form 𝑓 of π‘₯ is equal to π‘Ž will either be positive, negative, or equal to zero. If our value of π‘Ž is positive, the function will be positive. When π‘Ž is negative, it will be negative. And if π‘Ž is equal to zero, the function will be equal to zero. A linear function of the form 𝑓 of π‘₯ is equal to π‘šπ‘₯ plus 𝑏 will be positive, negative, and equal to zero for different values of π‘₯. We can find the value where it is equal to zero by setting 𝑓 of π‘₯ equal to zero. It is then useful to draw the graph of the function to identify the points where it is negative and positive.

A quadratic function of the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐 is usually positive, negative, and equal to zero for different values of π‘₯. To work out the values where it is equal to zero, we once again set the function equal to zero and then factor the equation to calculate our values. Once we’ve calculated these values, we can sketch the graph to find the points where the function is positive and negative. Whilst we didn’t see any questions of this type in the video, it is possible that the function has no values where it is equal to zero. In this case, the function would always be positive or always negative, as shown in the diagrams.

For the vast majority of questions we see, however, there will be solutions where 𝑓 of π‘₯ is equal to zero. If we can’t factor the function, we may still be able to solve it using the quadratic formula. We also saw that we can leave our answers to these type of questions using inequality signs or interval notation.

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