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 𝑎. As the line is below the 𝑥-axis,
this function will always be negative. Our second graph is a vertical
line, this time of the form 𝑥 equals some constant 𝑎. This function will be equal to zero
at the point the line crosses the 𝑥-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 𝑚𝑥 plus 𝑏, where 𝑚 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 𝑥-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 𝑥.
Let’s now consider what happens
when we have a quadratic or cubic function. A quadratic function of the form
𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 will be either u-shaped or n-shaped depending on the
sign of the leading coefficient 𝑎. We will once again be able to
calculate the values where the function is equal to zero by setting 𝑎𝑥 squared
plus 𝑏𝑥 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 𝑥 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 𝑥 equals negative eight looks like. If we consider the normal
coordinate axes, our horizontal axis is the 𝑥-axis and our vertical one is the
𝑦-axis. The 𝑦-axis can be replaced in this
case with 𝑓 of 𝑥. We are told that 𝑓 of 𝑥 is equal
to negative eight. Therefore, we need to find negative
eight on the 𝑦- or 𝑓 of 𝑥 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
𝑥-axis, it will always be negative. As 𝑓 of 𝑥 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 𝑚𝑥 plus 𝑏.
Determine the sign of the function
𝑓 of 𝑥 is equal to negative five 𝑥 plus five.
We know that this function is
linear as it is of the form 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚 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 𝑥-axis at one point. This will be the point where the
function is equal to zero. When the graph is above the
𝑥-axis, the function will be positive, and when it is below the 𝑥-axis, it will be
negative.
To calculate the point at which the
function is equal to zero, we will set 𝑓 of 𝑥 equal to zero. Adding five 𝑥 to both sides of
this equation gives us five 𝑥 is equal to five. We can then divide both sides of
this equation by five, giving us 𝑥 is equal to one. Our function is positive for all
𝑥-values less than one. The function is negative or below
the 𝑥-axis for all 𝑥-values greater than one. We can therefore conclude the
following. The function is positive when 𝑥 is
less than one, the function is negative when 𝑥 is greater than one, and, finally,
the function equals zero when 𝑥 equals one. The function 𝑓 of 𝑥 equals
negative five 𝑥 plus five is positive, negative, and equals zero for different
values of 𝑥.
In our next question, we will look
at a quadratic function.
Determine the sign of the function
𝑓 of 𝑥 is equal to 𝑥 squared plus 10𝑥 plus 16.
This function is quadratic and as
the coefficient of 𝑥 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 𝑥 is positive, negative, and
also equal to zero. We can begin by calculating the
zeroes of the function by setting 𝑓 of 𝑥 equal to zero. This gives us 𝑥 squared plus 10𝑥
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 𝑥 as 𝑥 multiplied by 𝑥 is 𝑥 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. 𝑥 squared plus 10𝑥 plus 16
factorized is equal to 𝑥 plus eight multiplied by 𝑥 plus two.
As multiplying these two
parentheses gives us zero, one of the parentheses themselves must also be equal to
zero. Either 𝑥 plus eight equals zero or
𝑥 plus two equals zero. Subtracting eight from both sides
of the first equation gives us 𝑥 equals negative eight. Subtracting two from both sides of
the second equation gives us 𝑥 is equal to negative two. This means that the function 𝑓 of
𝑥 equal to 𝑥 squared plus 10𝑥 plus 16 is equal to zero when 𝑥 equals negative
eight or 𝑥 equals negative two.
It is now worth sketching the graph
𝑦 equals 𝑥 squared plus 10𝑥 plus 16. We know that the graph is u-shaped
and crosses the 𝑥-axis when 𝑥 equals negative eight and when 𝑥 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 𝑥-axis. This occurs between the values
negative eight and negative two. We can write this as an
inequality. The function is negative when 𝑥 is
greater than negative eight, but less than negative two. The graph is positive when it is
above the 𝑥-axis. This occurs when 𝑥 is less than
negative eight or when 𝑥 is greater than negative two. We can now write all of this
information using interval and set notation.
The function 𝑓 of 𝑥 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 𝑥
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
𝑥 exists in the set of numbers negative eight, negative two. This means that it equals zero only
at the two values 𝑥 equals negative eight and 𝑥 equals negative two. We can therefore see that the
function 𝑓 of 𝑥 is equal to 𝑥 squared plus 10𝑥 plus 16 is positive, negative,
and equal zero for different values of 𝑥.
In our final question, we will
consider two different functions.
What are the values of 𝑥 for which
the functions 𝑓 of 𝑥 is equal to 𝑥 minus five and 𝑔 of 𝑥 is equal to 𝑥 squared
plus two 𝑥 minus 48 are both positive?
Let’s begin by considering the
function 𝑓 of 𝑥 is equal to 𝑥 minus five. If we want this to be positive, 𝑓
of 𝑥 must be greater than zero. This gives us 𝑥 minus five is
greater than zero. Adding five to both sides of this
inequality gives us 𝑥 is greater than five. 𝑓 of 𝑥 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 𝑥. This gives us 𝑥 squared plus two
𝑥 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. 𝑥 squared plus two 𝑥 minus 48
equals zero.
This can be factored or factorized
into two sets of parentheses or brackets. The first term in each bracket is
𝑥. 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 𝑥
minus six and 𝑥 plus eight. As the product of these two terms
is equal to zero, either 𝑥 minus six equals zero or 𝑥 plus eight is equal to
zero. Adding six to both sides of the
first equation gives us 𝑥 is equal to six. And subtracting eight from both
sides of the second equation gives us 𝑥 is equal to negative eight. This means that the function 𝑔 of
𝑥 is equal to zero when 𝑥 equals six and 𝑥 equals negative eight.
As our function is quadratic and
the coefficient of 𝑥 squared is positive, the graph will be u-shaped. This means that it is positive on
two sections, when 𝑥 is greater than six and when 𝑥 is less than negative
eight. The solution to the inequality 𝑥
squared plus two 𝑥 minus 48 is greater than nought is 𝑥 is less than negative
eight or 𝑥 is greater than six. This can also be written using
interval notation. 𝑔 of 𝑥 is positive in the open
interval negative ∞ to negative eight or the open interval six to ∞.
We want to work out the values of
𝑥 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 𝑥 is positive
for all values greater than five. 𝑔 of 𝑥 is positive for all values
less than negative eight and greater than six. This means that both functions are
positive when 𝑥 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 𝑥 is equal to 𝑎 will either be positive, negative, or equal to zero. If our value of 𝑎 is positive, the
function will be positive. When 𝑎 is negative, it will be
negative. And if 𝑎 is equal to zero, the
function will be equal to zero. A linear function of the form 𝑓 of
𝑥 is equal to 𝑚𝑥 plus 𝑏 will be positive, negative, and equal to zero for
different values of 𝑥. We can find the value where it is
equal to zero by setting 𝑓 of 𝑥 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
𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is usually positive, negative, and equal to zero for
different values of 𝑥. 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 𝑥 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.
