Question Video: Finding the Value of the Variation Function of a Quadratic Function Mathematics • Second Year of Secondary School

Video Transcript

If 𝑉 is the variation function for 𝑓 of 𝑥 is equal to 𝑥 squared minus four 𝑥 plus two, what is 𝑉 of negative 0.2 when 𝑥 is equal to eight?

We’re given a quadratic function 𝑓 of 𝑥 is 𝑥 squared minus four 𝑥 plus two and asked to find the value of the variation function if 𝑥 changes from 𝑥 is equal to eight by an amount of negative 0.2. The first thing we need to do then is to find the variation function 𝑉 of ℎ. We can do this using the definition for a function 𝑓 of 𝑥 where its variation function at 𝑥 is equal to 𝑎 is given by 𝑉 of ℎ is 𝑓 of 𝑎 plus ℎ minus 𝑓 of 𝑎. And that’s where ℎ is the change in 𝑥. In our case, our function 𝑓 of 𝑥 is 𝑥 squared minus four 𝑥 plus two. Our given value of 𝑥 is eight so that 𝑎 is equal to eight so that our variation function 𝑉 of ℎ is 𝑓 of eight plus ℎ minus 𝑓 of eight.

There are two ways we could approach this problem. We know that our change in 𝑥 is negative 0.2, and this means that ℎ is negative 0.2. And we could substitute this directly into our equation for 𝑉 of ℎ. Alternatively, we could first find 𝑉 of ℎ, the variation function in terms of ℎ, and then substitute ℎ is negative 0.2. We’re going to use the second method to find the variation function 𝑉 of ℎ. And to do this, we’re first going to substitute 𝑥 is equal to eight plus ℎ into our function 𝑓 of 𝑥. And this gives us eight plus ℎ squared minus four times 𝑎 plus ℎ plus two. Distributing our parentheses, this gives us 64 plus 16ℎ plus ℎ squared minus 32 minus four ℎ plus two. And collecting like terms, this gives us eight squared plus 12ℎ plus 34.

Now, to find 𝑓 of eight, we substitute 𝑥 equal to eight into our function 𝑓 of 𝑥, which gives us eight squared minus four times eight plus two. And this evaluates to 34. Now, with these two results into our function 𝑉 of ℎ, we have eight squared plus 12ℎ plus 34 minus 34, that is, ℎ squared plus 12ℎ. And this is our variation function 𝑉 of ℎ for 𝑓 of 𝑥 at 𝑥 is equal to eight. Now, to find 𝑉 of negative 0.2, we substitute negative 0.2 in place of ℎ. This gives us negative 0.2 squared plus 12 times negative 0.2. That is 0.04 minus 2.4, which is negative 2.36. 𝑉 of negative 0.2 for 𝑓 of 𝑥 is equal to 𝑥 squared minus four 𝑥 plus two when 𝑥 is equal to eight is therefore negative 2.36.