Question

The function f(x)=7-4+x^2 written in vertex form is f(x)=(x-2)^2+3 what is the axis of symmetry for he function

Answer 1

The axis of symmetry is x = 2.

The axis of symmetry can always be found in a quadratic by looking at the x value of the vertex. The beauty of vertex form is that it allows you to see that value right away.

In the vertex for f(x) = (x - h)^2 + k, the vertex is (h, k)

By looking at the vertex form given f(x) = (x - 2)^2 + 3, we can see that 2 has been put in place for h. This gives us our axis of symmetry.

Answer 2

When you complete the square from f(x) = 7 - 4x + x^2 to f(x) = (x - 2)^2 + 3, whatever is inside the brackets (x - 2) in this case, is related to the axis of symmetry.

The axis of symmetry is the line running through the vertex (minimum in this case)

The minimum is

x - 2 = 0

x = 2

The y value is the 3 outside the brackets. The minimum = (2,3)

The axis of symmetry is x = 2 which is the x value of the minimum.

Below is a graph to show you that theses are one and the same.

y = (x - 2)^2 + 3

and

y = x^2 - 4x + 7

The graph cannot be quite shown that way because you will get 2 graphs right on top of one another (because they are the same). I have drawn tow other graphs to indicate that the one you want is in the middle.

y = 7 - 4x + x^2 is the red graph

The blue graph is y = (x - 2)^2 + 2.5

The green graph is y = (x - 2)^2 + 3.5. You can see if you put in y = (x - 2)^2 + 3 you will get exactly the original red graph.