1. The problem asks to write an absolute value equation to determine the maximum and minimum circumference $x$ of a regulation discus, given the nominal circumference 691.15 mm and a manufacturing tolerance of 3.15 mm. 2. The absolute value equation for tolerance problems is generally written as: $$|x - \text{nominal}| \leq \text{tolerance}$$ where $x$ is the actual measurement. 3. Applying the values: $$|x - 691.15| \leq 3.15$$ This means the circumference $x$ can vary up to 3.15 mm above or below 691.15 mm. 4. To find the minimum and maximum values of $x$, solve the inequality: $$-3.15 \leq x - 691.15 \leq 3.15$$ 5. Add 691.15 to all parts: $$-3.15 + 691.15 \leq x \leq 3.15 + 691.15$$ 6. Simplify: $$688 \leq x \leq 694.3$$ So, the circumference $x$ must be between 688 mm and 694.3 mm. --- For the function $f(x) = -\frac{1}{2} |x - 4|$: 7. The table values are given for $x = -6, -3, -2, 0, 3, 5, 6, 9$ with corresponding $y$ values. 8. The key features of the function are: - $x$-intercept: 4 (where $f(x) = 0$) - $y$-intercept: $f(0) = -\frac{1}{2} |0 - 4| = -\frac{1}{2} \times 4 = -2$ - Vertex: $(4, 0)$ (minimum point since the graph opens downward) - Axis of symmetry: $x = 4$ - Domain: all real numbers - Range: $(-\infty, 0]$ because the function is always less than or equal to zero - Increasing on $(-\infty, 4)$ - Decreasing on $(4, \infty)$ - The function is negative or zero (never positive)