1. The problem is to sketch the graph of the function $$y = |3x| + 2$$ using symmetry and intercepts. 2. Identify key features: - The absolute value function $$|3x|$$ is V-shaped and symmetric about the y-axis because it depends on the absolute value of $$x$$. - The graph is shifted vertically up by $$2$$ units, so the vertex is at $$(0, 2)$$. 3. Find intercepts: - **y-intercept:** When $$x=0$$, $$y = |3 imes 0| + 2 = 0 + 2 = 2$$, so the graph crosses the y-axis at $$(0, 2)$$. - **x-intercept:** Solve $$0 = |3x| + 2$$ which is impossible because $$|3x| \\geq 0$$ and $$|3x| + 2 \\geq 2$$, so there are no x-intercepts. 4. Use symmetry and slopes of linear pieces: - For $$x > 0$$, $$y = 3x + 2$$ because $$|3x| = 3x$$ when $$x > 0$$. - For $$x < 0$$, $$y = -3x + 2$$ because $$|3x| = -3x$$ when $$x < 0$$. - The graph forms a "V" with vertex at $$(0, 2)$$. 5. Summarize: - The graph is V-shaped, vertex at $$(0, 2)$$. - It slopes upward with slope $$3$$ for $$x > 0$$ and slope $$-3$$ for $$x < 0$$. - No x-intercepts; y-intercept at $$(0, 2)$$. Final answer: The graph is a V-shaped plot with vertex at $$(0, 2)$$, symmetric about the y-axis, intersecting the y-axis at $$(0, 2)$$, and having linear arms with slopes $$3$$ and $$-3$$ on the right and left of the vertex respectively.