Let's complete the square. To do that, we will consider the variable terms on the function's right-hand side.
y= x^2+7x-8
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 of 7x.
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 3.5* 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 3.5^2 to complete the square. To keep the equation true, this term must be added to both sides.
y+ 3.5^2= x^2+2(3.5x)+ 3.5^2-8
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)
By rewriting our function so that it matches the graphing form exactly, we can find its vertex.
Function:& y=(x-( -3.5))^2+( -20.25)
Vertex:& ( -3.5, -20.25)
