Video: Identifying Vertex Form of a Quadratic Expression

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.