For the function f(x)=-4x+9 , evaluate and simplify the difference quotient.

To evaluate and simplify the difference quotient for the function $f(x)=-4x+9$, we need to follow these steps:<br /><br />1. Find the expression for $f(x+h)$ by substituting $x+h$ into the function.<br />2. Calculate the difference quotient $\frac{f(x+h)-f(x)}{h}$.<br />3. Simplify the resulting expression.<br /><br />Step 1: Find $f(x+h)$<br />Substitute $x+h$ into the function $f(x)=-4x+9$:<br />$f(x+h)=-4(x+h)+9$<br />$f(x+h)=-4x-4h+9$<br /><br />Step 2: Calculate the difference quotient<br />The difference quotient is given by $\frac{f(x+h)-f(x)}{h}$.<br />Substitute $f(x+h)=-4x-4h+9$ and $f(x)=-4x+9$ into the difference quotient:<br />$\frac{f(x+h)-f(x)}{h}=\frac{(-4x-4h+9)-(-4x+9)}{h}$<br />$\frac{f(x+h)-f(x)}{h}=\frac{-4h}{h}$<br /><br />Step 3: Simplify the expression<br />Simplify the expression $\frac{-4h}{h}$:<br />$\frac{-4h}{h}=-4$<br /><br />Therefore, the simplified difference quotient for the function $f(x)=-4x+9$ is $\boxed{-4}$.