1. **State the problem:** We need to graph the function $$y = -\left| \frac{1}{4}x \right| + 2$$ by transforming the parent absolute value function $$y = |x|$$. 2. **Recall the parent function and transformations:** The parent function is $$y = |x|$$, which is a V-shaped graph with vertex at the origin (0,0). 3. **Identify transformations:** - Inside the absolute value, $$\frac{1}{4}x$$ compresses the graph horizontally by a factor of 4. - The negative sign outside reflects the graph vertically across the x-axis. - The $$+2$$ shifts the graph vertically upward by 2 units. 4. **Write the transformed function:** $$y = -\left| \frac{1}{4}x \right| + 2$$ 5. **Find the vertex:** At $$x=0$$, $$y = -|0| + 2 = 2$$, so the vertex is at $$(0,2)$$. 6. **Find points on the graph:** Choose values for $$x$$ and calculate $$y$$: - For $$x=4$$: $$y = -\left| \frac{1}{4} \times 4 \right| + 2 = -|1| + 2 = -1 + 2 = 1$$ - For $$x=-4$$: $$y = -\left| \frac{1}{4} \times (-4) \right| + 2 = -| -1 | + 2 = -1 + 2 = 1$$ - For $$x=8$$: $$y = -\left| \frac{1}{4} \times 8 \right| + 2 = -|2| + 2 = -2 + 2 = 0$$ - For $$x=-8$$: $$y = -\left| \frac{1}{4} \times (-8) \right| + 2 = -| -2 | + 2 = -2 + 2 = 0$$ 7. **Describe the graph:** The graph is a V-shape with vertex at $$(0,2)$$, opening downward because of the negative sign. The arms decrease linearly with slope $$-\frac{1}{4}$$ on each side. 8. **Summary:** The graph is the absolute value graph compressed horizontally by 4, reflected vertically, and shifted up by 2.