Video Transcript
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.
