1. **State the problem:** Graph the function $$y = |\log_2(3 - \frac{1}{2}x)| + 5$$ and show each transformation step by step. 2. **Recall the base function:** The base function is $$y = \log_2(x)$$ which has a vertical asymptote at $$x=0$$ and passes through $$(1,0)$$. 3. **Analyze the inside of the logarithm:** The argument is $$3 - \frac{1}{2}x$$. 4. **Find the domain:** Set $$3 - \frac{1}{2}x > 0$$ to keep the log defined. $$3 > \frac{1}{2}x$$ $$\Rightarrow x < 6$$ So the domain is $$(-\infty, 6)$$. 5. **Rewrite the argument to see transformations:** $$3 - \frac{1}{2}x = -\frac{1}{2}x + 3$$ This is a horizontal reflection and scaling of $$x$$, plus a horizontal shift. 6. **Find the horizontal shift:** Solve $$3 - \frac{1}{2}x = 1$$ (since $$\log_2(1) = 0$$ is the x-intercept of the base log function). $$3 - \frac{1}{2}x = 1$$ $$\Rightarrow -\frac{1}{2}x = -2$$ $$\Rightarrow x = 4$$ So the graph crosses the x-axis at $$x=4$$. 7. **Transformations inside the log:** - Horizontal reflection about the y-axis because of the negative coefficient. - Horizontal scaling by factor $$2$$ (since coefficient is $$-\frac{1}{2}$$). - Horizontal shift right by $$4$$ units. 8. **Apply the absolute value:** The output of the log is reflected above the x-axis, so all negative values become positive. 9. **Vertical shift:** The entire graph is shifted up by $$5$$ units. 10. **Summary of transformations:** - Start with $$y = \log_2(x)$$. - Reflect horizontally and scale: $$x \to 3 - \frac{1}{2}x$$. - Take absolute value: $$y = |\log_2(3 - \frac{1}{2}x)|$$. - Shift vertically up by 5: $$y = |\log_2(3 - \frac{1}{2}x)| + 5$$. 11. **Graph shape:** - Vertical asymptote at $$x=6$$. - The graph passes through $$(4,5)$$ because $$\log_2(1) = 0$$ and absolute value does not change zero. - For $$x < 4$$, $$\log_2(3 - \frac{1}{2}x) > 0$$ so graph is above 5. - For $$4 < x < 6$$, $$\log_2(3 - \frac{1}{2}x) < 0$$ but absolute value reflects it above 5. 12. **Final function for graphing:** $$y = |\log_2(3 - \frac{1}{2}x)| + 5$$. This completes the transformations and explanation.