Let's complete the square. To do that, we will consider the variable terms on the function's right-hand side.
y= x^2-5x+7
To visualize how we complete the square, we draw a generic rectangle where the upper left corner has an area of x^2 and the adjacent rectangles each have an area that is half that of -5x.
Since the upper left corner has an area of x^2, it must be a square with a side length of x. This allows us to factor the adjacent rectangle's areas to - 2.5* x. With this information, we can also determine the area of the lower right rectangle which completes the square.
As we can see, we need to add (-2.5)^2 to complete the square. To keep the equation true, this term must be added to both sides.
y+ (-2.5)^2 = x^2+2(-2.5x)+ (-2.5)^2+7
Let's isolate f(x) and simplify.
When a quadratic function is written in graphing form, we can identify its vertex.
Graphing Form:& y=a(x- h)^2+ k
Vertex:& ( h, k)
Let's identify the vertex in our function.
Function:& y=(x- 2.5)^2+ 0.75
Vertex:& ( 2.5, 0.75)
