Video Transcript
Determine the variation function 𝑣 of ℎ for 𝑓 of 𝑥 is equal to negative eight 𝑥 squared minus five 𝑥 minus eight at 𝑥 is equal to negative one.
In this question, we’re asked to determine the variation function 𝑣 of ℎ for a given quadratic function 𝑓 of 𝑥 at a value of 𝑥 is equal to negative one. And to do this, let’s start by recalling what we mean by the variation function of a given function 𝑓 of 𝑥 at a particular value of 𝑥.
We know that the variation function 𝑣 of ℎ for a function 𝑓 of 𝑥 at a value of 𝑥 is equal to 𝑎 is given by 𝑓 evaluated at 𝑎 plus ℎ minus 𝑓 evaluated at 𝑎. And in our case, our function 𝑓 of 𝑥 is the quadratic function negative eight 𝑥 squared minus five 𝑥 minus eight and our value of 𝑎 is negative one. So we want to substitute this function and this value of 𝑎 into a variation function. And it’s worth noting the variation function is a measure of how much the function changes when 𝑥 changes from 𝑎 to 𝑎 plus ℎ.
So let’s start by substituting 𝑎 is negative one into our variation function. We get that 𝑣 of ℎ is equal to 𝑓 evaluated at negative one plus ℎ minus 𝑓 evaluated at negative one. Therefore, to determine our variation function, we’re going to need to evaluate our quadratic 𝑓 at negative one plus ℎ and at negative one. We do this by substituting these values into our quadratic.
So let’s start by evaluating 𝑓 at negative one plus ℎ. We get negative eight multiplied by negative one plus ℎ squared minus five times negative one plus ℎ minus eight. We then need to subtract the value of 𝑓 evaluated at negative one. So we substitute 𝑥 is equal to negative one into our function 𝑓 of 𝑥 and subtract the resulting value. We’re subtracting negative eight times negative one squared minus five times negative one minus eight.
Now, let’s simplify this expression. First, in the first term, we’re going to distribute the exponent over the parentheses. We can do this by using the binomial formula or by using the FOIL method. In either case, we get negative eight multiplied by one minus two ℎ plus ℎ squared. In our next term, we’re going to distribute negative five over the parentheses. We have negative five multiplied by negative one is equal to positive five and negative five multiplied by ℎ is negative five ℎ.
We then need to subtract eight from this expression, and we can evaluate the final part of this expression. Negative eight multiplied by negative one squared is negative eight. Negative five multiplied by negative one is positive five. And we need to subtract eight from this value. However, don’t forget we’re subtracting this expression. So we need to switch the sign of each of these three terms. We add eight, subtract five, and add eight, although it’s worth noting we could’ve combined these terms inside the parentheses and then just switched the sign. Either method would work.
We can now simplify this expression slightly. First, five minus five is equal to zero, and negative eight plus eight is equal to zero. We can now distribute negative eight over our parentheses. We just multiply each term inside the parentheses by negative eight. We get negative eight plus 16ℎ minus eight ℎ squared. And don’t forget we also need to subtract five ℎ and add eight to this value. So we have negative eight plus 16ℎ minus eight ℎ squared minus five ℎ plus eight.
We now want to collect like terms. First, we notice that negative eight plus eight is equal to zero. We can also notice we have 16ℎ minus five ℎ, and 16 minus five is equal to 11. So we can combine these two terms to get 11ℎ. And we’ll reorder our terms to be in descending powers of ℎ. And we can’t simplify this any further. So this gives us our final answer.
The variation function 𝑣 of ℎ for 𝑓 of 𝑥 is equal to negative eight 𝑥 squared minus five 𝑥 minus eight at 𝑥 is equal to negative one is 𝑣 of ℎ is equal to negative eight ℎ squared plus 11ℎ.
