What is the axis of symmetry of the graph of the function 𝑓 of 𝑥 equals 𝑥 plus three squared plus four?
This function is in vertex form. Vertex form is 𝑓 of 𝑥 equals 𝑥 minus ℎ squared plus 𝑘, where ℎ 𝑘 is the vertex and 𝑥 equals ℎ is the axis of symmetry. So let’s figure out what ℎ and 𝑘 are. So here is our function. And here is the vertex form. So if vertex form has 𝑥 minus ℎ, and we have 𝑥 plus three in our function, how would we have gotten a three to be positive?
So if we want this to turn into plus three, we would have to plug in a negative three, because 𝑥 minus negative three would be equal to 𝑥 plus three. So ℎ is equal to negative three.
Now let’s solve for 𝑘. In the vertex form, it’s plus 𝑘. And in our function, we have plus four. So 𝑘 is equal to four. So our vertex must be at negative three, four. And our axis of symmetry is 𝑥 equals ℎ. So the axis of symmetry would be 𝑥 equals negative three.
Let’s go ahead and try to sketch this graph of the function. We know that our vertex is at negative three, four. And because of the square in our function, this will be a parabola. So our graph would look something like this, because from our vertex, if we would go right one, we will need to go up one, because one squared is one. And then from our vertex again, if you would go right two, two squared is four, so we’d go up four. And then from our vertex, if we go left one, we must go up one because negative one squared is one. And from our vertex again, if we would go left two, we would need to go up four because negative two squared is four.
So here would be our axis of symmetry at 𝑥 equals negative three, because we could take our function and fold it over this line and it would be symmetrical. So once again, the axis of symmetry of the graph of this function will be 𝑥 equals negative three.