college algebra word problem

Question

Let f(x) = −x^² + 4x − 5.

 Find and simplify the difference quotient, f(x+h)-f(x)/h

Answer 1

f(x+h) = -(x+h)^2 +4(x+h) -5 = -x^2 -2xh - h^2 +4x +4h

f(x+h)-f(x) = -x^2 -2xh -h^2 +4x + 4h - (-x^2+4x-5)

= -x^2-2xh -h^2 +4x+4h +x^2 -4x +5

the 5's cancel, the x^2's cancel, the 4x's cancel, leaving

f(x+h)-f(x) =-2xh-h^2 +4h

what's interesting about this end result is that's it's divisible by h

[f(x+h)-f(x)]/h =- 2x -h +4

then you can set h=0 to get -2x+4 which is the first derivative of the original quadratic function

the limit as h goes to zero of [f(x+h)-f(x)]/h = -2x+4

Graph the quadratic and you get a downward opening parabola

with vertex or maximum point where -2x+4 = 0, or x=2, y=-4+8-5 = -1, (2,-1) is the vertex & maximum point on the parabola, where the slope = 0