If $f(x)$ is a linear function, $f(-4)=3$, and $f(4)=-1$, find an equation for $f(x)$

Step 1 ：We are given a linear function $f(x)$, and two points on the line, $(-4, 3)$ and $(4, -1)$.

Step 2 ：A linear function is of the form $f(x) = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

Step 3 ：We can find the slope $m$ using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substituting the given points into this formula, we get $m = \frac{-1 - 3}{4 - (-4)} = -0.5$.

Step 4 ：Now that we have the slope, we can find the y-intercept $c$ by substituting one of the points and the slope into the equation $y = mx + c$. Using the point $(-4, 3)$, we get $3 = -0.5(-4) + c$, which simplifies to $c = 1.0$.

Step 5 ：Substituting the values of $m$ and $c$ into the equation $f(x) = mx + c$, we get the equation of the line as $f(x) = -0.5x + 1$.

Step 6 ：\(\boxed{f(x) = -0.5x + 1}\) is the final answer.