y = 5 + 16x - 2x^2

Question

Finding the vertex of, y = 5 + 16x - 2x^2

Answer 1

x value of vertex of every parabola (quadratic function) having equation y =  f(x) = ax² + bx + c is at "h" = -b/(2a).

 

Then "k" is easily computed by substituting in h for x in the y-formula (i.e. f(h) = k).

 

Yet another method!

Answer 2

Richard,

 

I'm not really clear what level of math you are taking, but I'll take a stab at it and say that you're in Algebra 2.  The vertex will either be at the highest point or lowest point on this function, depending on whether it is "concave down"/"open down" or "concave up"/"open up."  The slope right at this point will be zero.  What might be less obvious is the symmetry of the graphed function on either side of the vertex.

 

You should be familiar with the quadratic equation which usually is used to help you find either the roots of a quadratic equation, or an idea of the number of roots that could be possible.  Another use for this formula is it can quickly tell you about where the symmetry occurs: x =-b/(2a) and represents the equation for the line of symmetry.  The vertex of the parabola (if you're not sure why its a parabola, that's another explanation) will occur on this line of symmetry.

 

So you could approach this task by factoring, as was already explained, and remember which values to pull off for the coordinates of the vertex.  Alternatively you could find the x-coordinate of the vertex by using the first part of the quadratic formula: x = -b/2a as I suggested.  

x = -(16)/(2*(-2)) or x=4.  

 

Substituting this value into the original quadratic equation gives the y-coordinate for the vertex:

y = 5 + 16(4) - 2(4)²

y = 37

 

So the vertex is at (4, 37).

 

Either way works.  Just depends on your personal preference and particular skillset.  It's helpful to know both!

Answer 3

y = 5 + 16x - 2x²

y = -2x² + 16x + 5

y = -2(x² - 8x     ) + 5

y = -2(x² - 8x + 16) + 5 + 32      completion of square

y = -2(x - 4)² + 37