Question

Finding a difference quotient for a linear or quadratic function

Answer 1

`-4x-2h-3`

Explanations:

Given the function

`f(x)=-2x^2-3x+6`

We are to look for the difference quotient:

`(f(x+h)-f(x))/(h)`

Get f(x + h)

`\begin{gathered} f(x+h)=-2(x+h)^2-3(x+h)+6 \\ f(x+h)=-2(x^2+2xh+h^2)-3x-3h+6 \\ f(x+h)=-2x^2-4xh-2h^2-3x-3h+6 \\ \end{gathered}`

Given f(x) expressed as:

`f(x)=-2x^2-3x+6`

Substitute both functions into the difference quotient;

`\begin{gathered} (-2x^2-4xh-2h^2-3x-3h+6-(-2x^2-3x+6))/(h) \\ \frac{-\cancel{2x^2}-4xh-2h^2-\cancel{3x}-3h+\cancel{6}+\cancel{2x^2}+\cancel{3x}-\cancel{6}}{h} \\ (-4xh-2h^2-3h)/(h) \end{gathered}`

Factor out "h" from the result to have

`\begin{gathered} \frac{\cancel{h}(-4x-2h-3)}{\cancel{h}} \\ -4x-2h-3 \end{gathered}`

This shows that the simplified form of the expression is -4x - 2h - 3