1. **State the problem:** We are given the function $$f(x) = -2|x-7| + 8$$ and want to understand its behavior and graph. 2. **Recall the absolute value definition:** The absolute value function $$|x-7|$$ is defined as: $$ |x-7| = \begin{cases} 7 - x, & x < 7 \\ x - 7, & x \geq 7 \end{cases} $$ 3. **Apply the definition to rewrite $$f(x)$$:** For $$x < 7$$: $$ f(x) = -2(7 - x) + 8 = -2 \cdot 7 + 2x + 8 = 2x - 14 + 8 = 2x - 6 $$ For $$x \geq 7$$: $$ f(x) = -2(x - 7) + 8 = -2x + 14 + 8 = -2x + 22 $$ 4. **Piecewise form of $$f(x)$$:** $$ f(x) = \begin{cases} 2x - 6, & x < 7 \\ -2x + 22, & x \geq 7 \end{cases} $$ 5. **Interpretation:** - For $$x < 7$$, the function is a line with slope 2 and y-intercept -6. - For $$x \geq 7$$, the function is a line with slope -2 and y-intercept 22. - The graph is a "V" shape opening downward because of the negative coefficient in front of the absolute value. 6. **Find the vertex:** At $$x=7$$, both pieces meet: $$ f(7) = 2 \cdot 7 - 6 = 14 - 6 = 8$$ $$ f(7) = -2 \cdot 7 + 22 = -14 + 22 = 8$$ So the vertex is at $$ (7, 8) $$. 7. **Summary:** The function $$f(x) = -2|x-7| + 8$$ is a downward "V" shaped graph with vertex at $$ (7, 8) $$, increasing with slope 2 for $$x < 7$$ and decreasing with slope -2 for $$x \geq 7$$.