Find the equation of the axis of symmetry of the function 𝑓 of 𝑥 equals negative seven 𝑥 squared minus 14𝑥 plus five given that 𝑥 is in negative four to two.
When an equation is in the form of 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, which our function is, the equation for the axis of symmetry is equal to 𝑥 equals negative 𝑏 over two 𝑎. So for our equation, we need to take negative 𝑏 over two 𝑎. And for ours, 𝑏 is negative 14. And since we need negative 𝑏, we need negative negative 14 divided by two 𝑎. And 𝑎 is negative seven.
So on our denominator we have two times negative seven. So on the numerator, negative one times negative 14 is positive 14. And on the denominator, two times negative seven is negative 14. And
four  divided by negative 14 is negative one. So our equation for the axis of symmetry would be 𝑥 equals negative one. And negative one is indeed between negative four and positive two, which was a piece of information that we were given. So this is a good way to check.
Now another way to do this would be to take our function and put it in vertex form. So choosing this other route of putting it in vertex form, our first step would be to group the first two terms together and then take a GCF from it, a greatest common factor. And we want to take out a GCF that way the number in front of 𝑥 squared is a positive one.
Now our next step is to make this piece inside of the parentheses something squared. And in order to do that, we need to take 𝑏 divided by two and square it and then we will do something else. So right now 𝑏 is two. So first we need to simplify inside the parentheses, and two divided by two is one, and one squared is one.
Now in order to add a number into a function or an equation, we have to keep it balanced. So if we would add this one to the inside of the parentheses, which is right here, then we also have to subtract that number to this equation. Well technically that number really isn’t a one; it is a negative seven times one. So if we just added in a negative seven, we need to somehow subtract a negative seven from the equation.
So subtracting a negative seven really means we need to add a positive seven. So right now, the negative seven times one will mean a negative seven so we need to add a positive seven. And we can do so right on the end. So to simplify, the 𝑥 squared plus two 𝑥 plus one inside of the parentheses can simplify to be 𝑥 plus one squared because 𝑥 plus one times 𝑥 plus one is equal to 𝑥 squared plus two 𝑥 plus one.
And on the outside of parentheses, we have five plus seven, which is 12. So here is our equation in vertex form. And this is useful for a few reasons. Looking at our function in vertex form, the number on the outside of the parentheses tells us how far up or down to go. So 12 means our vertex will begin by going up 12, but now we have to know how far left or right. And that’s the number on the inside of the parentheses.
And the number on the inside of the parentheses, it will be opposite of what we see. So if that’s a positive one, that means we actually need to move negative one, so to the left one. Therefore, the vertex of this function would be negative one, 12. So it’s not only useful to find the vertex of the function, which is why it’s called the vertex form, but it also can tell us the axis of symmetry because of the axis of symmetry will go through the vertex.
So the axis of symmetry would be 𝑥 equals negative one. Now lastly, the fact that the seven is negative, so the number in front of the parentheses, since it’s negative, that means the parabola will be upside down. And multiplied by seven means it’s gonna become a little bit skinnier. So we do not have to worry too much about that right now, we can just sketch it. So our function will look something like this. So once again, the axis of symmetry of our function would be 𝑥 equals negative one.