Video Transcript
Determine the variation function 𝑉
of ℎ for 𝑓 of 𝑥 equals negative four 𝑥 squared minus nine 𝑥 plus nine at 𝑥
equals negative one.
In this question, we are given a
quadratic function 𝑓 of 𝑥 and asked to find its variation function at a given
value of 𝑥. To do this, we can start by
recalling that the variation function 𝑉 of ℎ of 𝑓 of 𝑥 at 𝑥 equals 𝑎 is given
by 𝑉 of ℎ equals 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎. In other words, we evaluate the
function at 𝑎 and then subtract this from the function evaluated at 𝑎 plus ℎ. The variation function 𝑉 of ℎ
tells us how much the function 𝑓 varies as the value of 𝑥 changes from 𝑎 to 𝑎
plus ℎ. For example, if 𝑉 of ℎ is
positive, then the function increases its output as the input changes from 𝑎 to 𝑎
plus ℎ. And if 𝑉 of ℎ is negative, then it
decreases its output.
We can use this definition to
determine the variation function of the given quadratic at 𝑥 equals negative
one. We have that 𝑓 of 𝑥 is negative
four 𝑥 squared minus nine 𝑥 plus nine and 𝑎 equals negative one. We can substitute 𝑎 equals
negative one into the formula for the variation function to obtain 𝑉 of ℎ equals 𝑓
of negative one plus ℎ minus 𝑓 of negative one. We can evaluate this in steps. Let’s start with 𝑓 of negative one
plus ℎ. We substitute this into the given
function 𝑓 of 𝑥 to get negative four times negative one plus ℎ all squared minus
nine times negative one plus ℎ plus nine.
We could follow the same process
for 𝑓 evaluated at negative one; however, we can just directly evaluate this for
simplicity. We substitute 𝑥 equals negative
one into the function 𝑓 to get negative four times negative one squared minus nine
times negative one plus nine. We can evaluate this to get 14. We can now substitute this into our
expression for 𝑉 of ℎ. We want to simplify this expression
for the variation function. We can do this in stages. First, we can expand the
exponent. We get one minus two ℎ plus ℎ
squared. We need to multiply this by
negative four.
Next, we can distribute negative
nine over the parentheses. This gives us nine minus nine
ℎ. We can also evaluate the final two
terms. We have nine minus 14 is equal to
negative five. This gives us the following
expression for the variation function. We can simplify further by
distributing the factor of negative four over the parentheses. This gives us negative four plus
eight ℎ minus four ℎ squared plus nine minus nine ℎ minus five.
We now want to combine the like
terms. We only have a single ℎ squared
term of negative four ℎ squared. We can combine the ℎ terms to
obtain negative ℎ. Finally, we have negative four plus
nine minus five equals zero. This then gives us the variation
function 𝑉 of ℎ of 𝑓 at 𝑥 equals negative one is negative four ℎ squared minus
ℎ.
