1. **State the problem:** We are given the black curve with equation $y = x^3 + 2x^2 + 1$. We need to: a) Write down an equation of the red curve. b) Describe the transformation that maps the black curve onto the red curve. 2. **Analyzing the graph description:** The red curve is a cubic-like curve positioned to the right of the y-axis, starting around $x=1$ and extends to $x=6$. It is described as a horizontal translation of the black curve to the right. 3. **Understanding horizontal translation:** If a function $y = f(x)$ is shifted horizontally by $h$ units to the right, the new function becomes $y = f(x - h)$. 4. **Writing the equation of the red curve:** Since the red curve is the black curve shifted to the right by approximately 1 unit, we replace $x$ by $x - 1$ in the original function: $$ y = (x - 1)^3 + 2(x - 1)^2 + 1 $$ 5. **Describing the transformation:** The transformation is a horizontal translation (shift) of the black curve 1 unit to the right. **Final answers:** a) Equation of the red curve: $$ y = (x - 1)^3 + 2(x - 1)^2 + 1 $$ b) The red curve is obtained by shifting the black curve 1 unit to the right horizontally.