1. The problem gives the black curve with equation $$y = x^3 + 2x^2 + 1$$ and asks for the equation and transformation of the red curve relative to the black curve. 2. Observing the red curve compared with the black one, it appears to be a vertical shift (translation) upward. 3. To find the exact translation, compare a specific point's y-values on both curves. For example, at $$x=0$$: - Black curve: $$y = 0^3 + 2\cdot0^2 + 1 = 1$$ - The red curve is higher; from the graph description, assume the red curve at $$x=0$$ is roughly at $$y=3$$. 4. The vertical shift $$k$$ can be found from: $$ y_{red} = y_{black} + k \implies 3 = 1 + k \implies k = 2 $$ 5. Therefore, the red curve's equation is: $$ y = x^3 + 2x^2 + 1 + 2 = x^3 + 2x^2 + 3 $$ 6. The transformation is a vertical translation upward by 2 units. Final answers: - a) Red curve equation: $$y = x^3 + 2x^2 + 3$$ - b) Transformation: shift the black curve vertically upward by 2 units.