How would I find a difference quotient for the formula y=-x^2+x-2
To find the difference quotient for the given formula y = -x^2 + x - 2, we need to find the expression for the slope of the secant line between two points on the graph of the function as the distance between the points approaches zero.
The difference quotient is given by the formula:
(f(x + h) - f(x)) / h
Let's calculate this step by step.
Step 1: Substitute x + h into the given function:
f(x + h) = -(x + h)^2 + (x + h) - 2
= -x^2 - 2hx - h^2 + x + h - 2
= -x^2 + x - 2 - 2hx - h^2 + h
Step 2: Substitute f(x + h) and f(x) into the difference quotient formula:
(f(x + h) - f(x)) / h = [(-x^2 + x - 2 - 2hx - h^2 + h) - (-x^2 + x - 2)] / h
= (-x^2 + x - 2 - 2hx - h^2 + h + x^2 - x + 2) / h
= (-2hx - h^2 + h) / h
= -2x - h + 1
Therefore, the difference quotient for the function y = -x^2 + x - 2 is -2x - h + 1.