Video Transcript
Which of the following is the
vertex form of the function 𝑓 of 𝑥 equals two 𝑥 squared plus 12𝑥 plus 11. And we’re given five answer
options.
So, we have a quadratic function in
its general or expanded form, and we’re asked to find its vertex form. That’s 𝑓 of 𝑥 equals 𝑎
multiplied by 𝑥 plus 𝑝 all squared plus 𝑞, where 𝑎, 𝑝, and 𝑞 are constants we
need to find. To answer to this question then, we
need to write our quadratic function in its completed square form.
Now, this is slightly trickier than
usual because the coefficient of 𝑥 squared in our function isn’t one. It is two. Now, we can deal with this by
factoring by this coefficient. You can either factor from all
three terms giving two multiplied by 𝑥 squared plus six 𝑥 plus 11 over two. Or we can simply factor this
coefficient from the first two terms, giving two multiplied by 𝑥 squared plus six
𝑥 plus 11. And personally, I think the second
method is easier.
What we’re now going to do is
complete the square on the expression within the parentheses. That’s 𝑥 squared plus six 𝑥. We begin by halving the coefficient
of 𝑥 to give the number inside the parentheses. So, we have 𝑥 plus three all
squared. And then, we need to subtract the
square of this value. So, we have 𝑥 plus three all
squared minus nine. A quick check of redistributing
these parentheses confirms that 𝑥 plus three all squared minus nine is indeed equal
to 𝑥 squared plus six 𝑥.
Now, we need to be very careful
here. All of this expression of 𝑥
squared plus six 𝑥 was being multiplied by two. So, we need to put a large set of
brackets or parentheses around 𝑥 plus three all squared minus nine, multiply it by
two. And then, we still have the 11 that
we were adding on. So, we found that our function 𝑓
of 𝑥 is equivalent to two multiplied by 𝑥 plus three all squared minus nine plus
11.
Next, we need to distribute the
two. So, we have two multiplied by 𝑥
plus three all squared and then two multiplied by negative nine, which is negative
18, and then plus 11. Remember that 11 is not being
multiplied by two. Finally, we just simplify negative
18 plus 11 is negative seven. So, we have our quadratic in its
vertex form two multiplied by 𝑥 plus three all squared minus seven.
Looking carefully at the five
answer options we were given because they’re all quite similar, we see that this is
answer option (a). Now, in this question, we just
worked through the completing-the-square process ourselves and then determined the
correct answer option. It would actually have been
possible to eliminate a couple of the options straightaway though. If we look at the vertex form of
the function, we see that within the parentheses we have just 𝑥 plus 𝑝 all
squared. We could, therefore, have
eliminated options (b) and (c) as they are not the vertex form of any function
because inside the parentheses they each have two 𝑥.
We could also have eliminated
option (e) because we can see that there is no two involved in this option. And as the coefficient of 𝑥
squared in our original function was two, we’d need a factor of two outside the
parentheses as we have in answer options (a) and (d). The only difference between options
(a) and (d) is the sign inside the parentheses. We have positive three for option
(a) and negative three for option (d). We should remember, though, that
the sign here is always the same as the sign of the coefficient of 𝑥 in the
original function. So, in this case, it’s
positive. In any case, we have our answer
though. The vertex form of this function is
𝑓 of 𝑥 equals two multiplied by 𝑥 plus three all squared minus seven.
