1. **State the problem:** We are given the linear function $y = -x + 4$ and a table of values for $x$ from $-3$ to $3$. We want to understand how to calculate $y$ for each $x$ and interpret the function. 2. **Formula and explanation:** The function is $y = -x + 4$. This means for any input $x$, multiply by $-1$ and then add $4$. 3. **Calculate values step-by-step:** - For $x = -3$: $y = -(-3) + 4 = 3 + 4 = 7$ - For $x = -2$: $y = -(-2) + 4 = 2 + 4 = 6$ - For $x = -1$: $y = -(-1) + 4 = 1 + 4 = 5$ - For $x = 0$: $y = -(0) + 4 = 0 + 4 = 4$ - For $x = 1$: $y = -(1) + 4 = -1 + 4 = 3$ - For $x = 2$: $y = -(2) + 4 = -2 + 4 = 2$ - For $x = 3$: $y = -(3) + 4 = -3 + 4 = 1$ 4. **Interpretation:** The function is a straight line with slope $-1$ and y-intercept $4$. The slope $-1$ means the line goes down one unit vertically for every one unit it moves right horizontally. 5. **Summary:** The completed table is: | $x$ | $y = -x + 4$ | |-----|--------------| | -3 | 7 | | -2 | 6 | | -1 | 5 | | 0 | 4 | | 1 | 3 | | 2 | 2 | | 3 | 1 | This confirms the linear relationship and the values of $y$ for each $x$.