For each quadratic function, find the vertex of its graph.
y= x^2 + 8x +16

Question

For each quadratic function, find the vertex of its graph.
y= x^2 + 8x +16

anonymous · Answer

The vertex of f(x)=ax^2+bx+c is (h,k)=(-b/2a, f(-b/2a)).  In this case,$$x=\frac{-b}{2a}=\frac{-8}{2}=-4$$$$f(-4)=0 \implies (h,k)=(-4,0)$$After a little closer inspection, it is clear this function factors to y=(x+4)^2, which means its single root is x=-4, which is also at the vertex.

anonymous · Answer

A juggler tosses a ring into the air. The height of the ring in feet above the juggler's hands can be modeled by the function 
f(x)= -16x^2 + 16x , where x is the time in seconds after the ring is tossed. Find the ring's maximum height above the juggler's hands and the time it takes the ring to reach the height. Then find how long the ring is in the air. 
y= -x^2 + 6x - 4

anonymous · Answer

Find the vertex as above.  The value of y=f(x) at the vertex is the maximum height.  Solve for the value of x (time) that makes f(x)=0 to find when the ring gets back to the juggler's hand.