1. The problem asks to analyze and sketch the graph of the function $$y = 21 (x - 3)^2 + 2$$ by translating, reflecting, compressing, and stretching the graph of the base function $$y = x^2$$. 2. The base function is $$y = x^2$$, which is a parabola opening upwards with vertex at the origin (0,0). 3. The given function is $$y = 21 (x - 3)^2 + 2$$. 4. Let's identify the transformations step-by-step: - The term $$(x - 3)^2$$ indicates a horizontal translation 3 units to the right. - The coefficient $$21$$ in front of the squared term means a vertical stretch by a factor of 21. - The constant $$+2$$ outside the squared term means a vertical translation 2 units upwards. 5. So, starting from $$y = x^2$$: - Shift the graph right by 3 units: vertex moves from (0,0) to (3,0). - Stretch vertically by 21: the parabola becomes much narrower. - Shift up by 2: vertex moves to (3,2). 6. The vertex form of a parabola is $$y = a(x - h)^2 + k$$ where $$(h,k)$$ is the vertex and $$a$$ controls the stretch/compression and direction. 7. Here, $$a = 21$$, $$h = 3$$, and $$k = 2$$. 8. The graph opens upwards because $$a > 0$$. 9. Final vertex: $$(3, 2)$$. 10. The graph is very narrow due to the large stretch factor 21. 11. To confirm, you can plot points around the vertex, for example: - At $$x=2$$, $$y = 21(2-3)^2 + 2 = 21(1)^2 + 2 = 23$$. - At $$x=4$$, $$y = 21(4-3)^2 + 2 = 21(1)^2 + 2 = 23$$. 12. This confirms the parabola is narrow and symmetric about $$x=3$$. Final answer: The graph of $$y = 21 (x - 3)^2 + 2$$ is the parabola $$y = x^2$$ shifted right 3 units, up 2 units, and vertically stretched by a factor of 21 with vertex at $$(3, 2)$$.