Ule 11.3 More on Functions and craphs \[ 1 \leftarrow \mid \text { Find the difference quotient of } f \text {, that is, find } \frac{f(x+h)-f(x)}{h}, h \neq 0 \text {, for the following function. Be sure to simplify. } \] \[ \frac{f(x)=x^{2}-8 x+2}{h}=\square \text { (Simplify your answer.) } \]

To find the difference quotient \(\frac{f(x+h) - f(x)}{h}\) for the function \(f(x) = x^2 - 8x + 2\), we start by calculating \(f(x+h)\): \[ f(x+h) = (x+h)^2 - 8(x+h) + 2 \] Expanding this, we get: \[ f(x+h) = (x^2 + 2xh + h^2) - (8x + 8h) + 2 \] Now, simplifying this gives: \[ f(x+h) = x^2 + 2xh + h^2 - 8x - 8h + 2 \] Now, substituting back into the difference quotient: \[ \frac{f(x+h) - f(x)}{h} = \frac{(x^2 + 2xh + h^2 - 8x - 8h + 2) - (x^2 - 8x + 2)}{h} \] This simplifies to: \[ \frac{2xh + h^2 - 8h}{h} \] Factoring out \(h\) gives us: \[ \frac{h(2x + h - 8)}{h} \] Since \(h \neq 0\), we can cancel \(h\): \[ 2x + h - 8 \] So the simplified difference quotient is: \[ \boxed{2x + h - 8} \] Now you have your difference quotient neatly wrapped up, ready for action!